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     {\huge СИБИРСКИЕ \ ЭЛЕКТРОННЫЕ} \\[2mm]  %Headings in Russian
     {\huge МАТЕМАТИЧЕСКИЕ ИЗВЕСТИЯ} \\[2mm]
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     {\flushleft\it Том 19, стр. 144--144 (2022) \hspace{\fill}{\rm\small УДК \udcs}} %Volume, pages in Russian,
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\begin{document}
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\title[Nonstandard quasivarieties]{On nonstandard quasivarieties of differential groupoids and unary algebras}
\author{A.~V. Kravchenko}
\address{Aleksandr Vladimirovich Kravchenko
\newline\hphantom{iii} Sobolev Institute of Mathematics,
\newline\hphantom{iii} pr. Koptyuga 4,
\newline\hphantom{iii} 630090 Novosibirsk, Russia;
\newline\hphantom{iii} Novosibirsk State University of Economics and Management,
\newline\hphantom{iii} ul. Kamenskaya 56,
\newline\hphantom{iii} 630099 Novosibirsk, Russia}
\email{a.v.kravchenko@mail.ru}%
\author{M.~V. Schwidefsky}
\address{Marina Vladimirovna Schwidefsky
\newline\hphantom{iii} Sobolev Institute of Mathematics,
\newline\hphantom{iii} pr. Koptyuga 4,
\newline\hphantom{iii} 630090 Novosibirsk, Russia}
\email{semenova@math.nsc.ru}%
%
\thanks{\sc Kravchenko, A.~M., Schwidefsky, M.~V., On nonstandard quasivarieties of differential groupoids and unary algebras}
\thanks{\copyright \ 2022 Kravchenko, A.~V., Schwidefsky, M.~V.}
\thanks{The research was carried out in the framework of the state contract of Sobolev Institute of Mathematics SB RAS,
project no. FWNF-2022-0012. The results of Section 3 were obtained under the support of the Russian Science Foundation, project number 22-21-00104.}
\thanks{\it Received  June 20, 2022, published  ????.}%


\semrtop \vspace{1cm}
\maketitle {\small
\begin{quote}
\noindent{\sc Abstract. }
We prove that there are continuum many nonstandard quasivarieties of differential groupoids and unary algebras.
\medskip

\noindent{\bf Keywords:} quasivariety, nonstandard quasivariety, differential group\-o\-id, unary algebra.
 \end{quote}
}

\section{Introduction}

A prevariety $\mathbf{K}$ is \emph{profinite\/} if each Boolean topological structure with its algebraic reduct belonging to $\mathbf{K}$ is \emph{profinite with respect to\/}~$\mathbf{K}$; that is, isomorphic
to an inverse limit of finite structures (equipped with the discrete topology) belonging to $\mathbf{K}$.
A universal Horn class $\mathbf{K}$ is \emph{standard\/} if each Boolean topological structure with its algebraic reduct in $\mathbf{K}$
is isomorphic to a closed substructure of the~Cartesian product of a~nonempty family of finite structures from $\mathbf{K}$, where all the finite structures are equipped with
the discrete topology and the~Cartesian product is equipped with the product topology.

The notion of a~standard class is closely related with natural dualities and profinite structures, see \cite{CD,Za}. A detailed study of standard and profinite classes of general algebraic structures as well as particular classes of algebraic structures was carried out
in \cite{CDF,CDH,DCJ,DJMM}. It was established in \cite{NS} that there is no algorithm which would decide if the variety generated by a given finite structure
is standard. This result solved in the negative a problem from~\cite{CDF,DCJ} in the~case of varieties. A wide spectrum of examples of standard and nonstandard quasivarieties can be found in \cite{BMN,DCJ,DJMM,T}.

In a~recent article of the~authors and their co-author, the~notion of a~B-class was introduced
which allows us to treat certain complexity problems for quasivarieties in a~uniform way~\cite{KrNuSchw18}. Namely, the existence
of such a~class with respect to a~quasiva\-ri\-ety implies extremely high level of complexity from many points
of view. In addition to known properties of $Q$-universality and the undecidability of certain decision problems,
the existence of a~B-class leads to continuum many
subquasivarieties lacking finitely partitionable quasi-equational bases, continuum many subquasivar\-ie\-ties
whose quasi-equational theory is undecidable, and contin\-u\-um many nonstan\-dard subquasivarieties~\cite{KrNuSchw19,KrNuSchw21}.

Although the variety $\mathbf{Dm}$ of differential groupoids and a certain quasivariety $\mathbf{V}$ of unary
algebras with two unary operations do not admit B-classes, a series of complexity results holds for them.
For example, the~representation, undecidability, and independent axiomatization results can be proven
for these two quasivarieties, see~\cite{DG2,DG}.
Notice that the~proof of the~facts that the~quasivarieties~$\mathbf{Dm}$ and~$\mathbf{V}$ are $Q$-universal
in~\cite{Kr08} and~\cite{Kr02} required an individual approach and adjustment of known sufficient conditions for
$Q$-universality.
In the~present article, we prove that, despite of the lack of $\mathrm{B}$-classes in $\mathbf{Dm}$ and~$\mathbf{V}$,
these quasivarieties also contain continuum many nonstandard subquasivarieties, see Theorems \ref{T:1dg}, \ref{T:2dg}, \ref{T:1ua}, and \ref{T:2ua}.

\section{Basic definitions and auxiliary results}

For all definitions and notation concerning (algebraic) structures and their quasi\-var\-ie\-ties, we refer to the~monograph~\cite[Ch.~1]{G} and the~articles~\cite{DG2,DG,KrNuSchw18,KrNuSchw19,KrNuSchw20,KrNuSchw21}.

\subsection{Quasivarieties and class operators}

{\it Quasi-identities\/} are universal Horn sentences of the form
$$
\forall\overline{x}\
[\varphi_1(\overline{x})\ \& \dots \&\ \varphi_k(\overline{x}) \longrightarrow \varphi_0(\overline{x})],
$$
where $\varphi_i(\overline{x})$ is an~atomic formula for each $i\leqslant k$.
A~class~$\mathbf{K}$ of structures is a~{\it quasivariety\/} if it coincides with the~class of all models of some set~$\Phi$ of quasi-identities.
Then the set~$\Phi$ is called a~{\it quasi-equational basis\/} of~$\mathbf{K}$.

We denote structures by calligraphic letters. The~universe of a~structure is denoted by the~corresponding italic letter.
For classes of structures, we use boldface letters. We assume that all classes are {\it abstract}, i.e., closed under isomorphism.

Let $\mathbf{K}(\sigma)$ denote the~class of all structures of type~$\sigma$.
For a class
$\mathbf{K}\subseteq\mathbf{K}(\sigma)$,
let $\mathbf{K}_{\rm{fin}}$ denote the class of finite structures from $\mathbf{K}$ and
let $\mathbf{Q}(\mathbf{K})$ denote the least quasivariety extending $\mathbf{K}$.
Let $\mathbf{H}(\mathbf{K})$ denote the class of homo\-mor\-phic images of structures from $\mathbf{K}$;
let $\mathbf{P}(\mathbf{K})$ denote the class of structures that are isomorphic to the Cartesian product of a family of structures from $\mathbf{K}$;
let $\mathbf{P}^+(\mathbf{K})$ denote the class of structures that are isomorphic to the Cartesian product of a nonempty family of structures from $\mathbf{K}$;
let $\mathbf{S}(\mathbf{K})$ denote the class of structures that are embeddable into structures from $\mathbf{K}$;
see, for example,~\cite[Sec.~1.2.7]{G}.
Finally, let $\mathbf{T}$ denote the trivial (quasi)variety.

We recall the notion of the~inverse (projective) limit, see, for example, \cite[Sec.~1]{DCJ}.

\begin{definition}\label{D:lim}
A triple $\Lambda=\langle I,\mathcal{A}_i,\pi_{ij}\rangle$ is an {\it inverse spectrum\/} if
$\langle I;\leqslant\rangle$ is an up-directed partially ordered set,
$\{\mathcal{A}_i\mid i\in I\}$ is a set of structures of the same similarity type $\sigma$,
the mapping $\pi_{ij}\colon\mathcal{A}_j\to\mathcal{A}_i$ is a homomorphism for all $i,j\in I$ with $i\leqslant j$,
and the following holds:
\begin{enumerate}
\item
the mapping $\pi_{ii}$ is the identity automorphism on $\mathcal{A}_i$ for each $i\in I$;
\item
we have $\pi_{ik}=\pi_{ij}\pi_{jk}$ for all $i,j,k\in I$ with $i\leqslant j\leqslant k$.
\end{enumerate}
If each homomorphism $\pi_{ij}$ is onto then the inverse spectrum is said to be {\it surjective}.
A structure $\mathcal{A}\in\mathbf{K}(\sigma)$ is the {\it inverse limit\/} of~$\Lambda$ if its universe
\[
A=\bigl\{(a_i\mid i\in I)\in\prod_{i\in I}A_i\mid\pi_{ij}(a_j)=a_i\ \text{for all}\ i\leqslant j\ \text{in}\ I\bigr\}
\]
is not empty.
\end{definition}

We denote the inverse limit of $\Lambda$ whenever it exists by $\varprojlim\Lambda$.
It is clear that $\varprojlim\Lambda\in\mathbf{S}\mathbf{P}(\mathcal{A}_i\mid i\in I)$ if $\Lambda$ is as in Definition \ref{D:lim}.
For $i\in I$, we denote by $\pi_i$ the canonical projection from $\varprojlim\Lambda$ to $\mathcal{A}_i$.
Notice that $\pi_i$ is surjective if so is~$\Lambda$.

A subclass $\mathbf{K}^\prime$ of $\mathbf{K}$ is a $\mathbf{K}$-{\it quasivariety\/}
if $\mathbf{K}^\prime$ is defined within $\mathbf{K}$ by a~set of quasi-identities
or, equivalently, $\mathbf{K}^\prime=\mathbf{Q}(\mathbf{K}^\prime)\cap\mathbf{K}$.

\subsection{Finite B-classes}

As usual, we denote the least infinite cardinal by $\omega$.
For a~set~$X$, we denote by $\mathcal{P}_{\mathrm{fin}}(X)$ the set of all finite subsets of $X$
and by $\mathcal{P}_{\mathrm{inf}}(X)$ the set of all infinite subsets $I\subseteq X$ such that the~complement $X{\setminus}I$ is infinite too.
Notice that $|\mathcal{P}_{\mathrm{inf}}(X)|=2^\omega$ for each countable infinite set $X$.

The following notion is introduced in~\cite{KrNuSchw18}.

\begin{definition}\label{D:B}
Let $\mathbf{M}\subseteq\mathbf{K}(\sigma)$ be a quasivariety of a~finite type~$\sigma$.
A class $\mathbf{A}=\{\mathcal{A}_X\mid X\in\mathcal{P}_{\mathrm{fin}}(\omega)\}\subseteq\mathbf{M}$ of finite structures is a {\it  finite} B-{\it class\/} with respect to $\mathbf{M}$ if $\mathbf{A}$ satisfies the following conditions:
\begin{enumerate}
\item[$(\mathrm{B}_0)$]
$\mathcal{A}_\varnothing$ is a trivial structure;
\item[$(\mathrm{B}_1)$]
if $X=Y\cup Z$ in $\mathcal{P}_{\mathrm{fin}}(\omega)$ then $\mathcal{A}_X\in\mathbf{Q}(\mathcal{A}_Y,\mathcal{A}_Z)$;
\item[$(\mathrm{B}_2)$]
if $\varnothing\ne X\in\mathcal{P}_{\mathrm{fin}}(\omega)$ and $\mathcal{A}_X\in\mathbf{Q}(\mathcal{A}_Y)$ then $X=Y$;
\item[$(\mathrm{B}_3)$]
if $F\in\mathcal{P}_{\mathrm{fin}}(\omega)$, $i\in\omega$, and $f\colon\mathcal{A}_F\to\mathcal{A}_{\{i\}}$ is a homomorphism
then either $f(\mathcal{A}_F)\cong\mathcal{A}_\varnothing$ or $i\in F$;
\item[$(\mathrm{B}_4)$]
if $F\in\mathcal{P}_{\mathrm{fin}}(\omega)$ then $\mathbf{H}(\mathcal{A}_F)\cap\mathbf{M}\subseteq\mathbf{A}$.
\end{enumerate}
\end{definition}

As is mentioned in the introduction, the existence of a (finite) B-class with respect to a quasivariety~$\mathbf{K}$
witnesses a very complicated structure of~$\mathbf{K}$ as well as of its quasivariety lattice. However,
there exists natural classes of structures that do not admit B-classes by obvious reasons (say, the~existence of
homomorphic images in ``small'' subvarieties, which spoils $\mathrm{B}_3$ or $\mathrm{B}_4$).
In the following two subsections, we recall necessary information about two such classes.

\subsection{Differential groupoids}
A~{\it differential groupoid\/} is an algebra endowed with one binary operation~$\cdot$
that satisfies the following identities:
\begin{gather*}
\forall x\  [x\cdot x=x],
\quad
\forall x\,\forall y\  [x\cdot(x\cdot y)=x],
\\
\forall x\,\forall y\,\forall z\,\forall t\
[(x\cdot y)\cdot(z\cdot t)=(x\cdot z)\cdot(y\cdot t)].
\end{gather*}
Let~$\mathbf{Dm}$ denote the~variety of all differential groupoids.

For brevity, we write $x_1 x_2\dots x_n$ for $(\dots(x_1\cdot x_2)\cdot \dots)\cdot x_n$ and $xy^n$ for $x \underbrace{y\dots y}_n$.
We use the~following representation of differential groupoids from~\cite{RoRo87}.
A~groupoid $\mathcal{G}$ is an~{\it $\mathbf{Lz}$-$\mathbf{Lz}$-sum\/}
(of orbits $\mathcal{G}_i$ over a~groupoid $\mathcal{I}$) {\it satisfying the~left normal law\/} if there is a~partition $G=\bigcup_{i\in I}G_i$ such that, for every pair $(i,j)\in I^2$, there is a~mapping $h^j_i\colon G_i\to G_i$ satisfying the~following conditions:
\begin{itemize}
\item[(i)] for every $i\in I$, $h^i_i$ is the~identity mapping;
\item[(ii)] we have $h^j_i(h^k_i(x))=h^k_i(h^j_i(x))$ for all $i,j,k\in I$ and $x\in G_i$;
\item[(iii)] we have $a_i\cdot a_j=h^j_i(a_i)$ for all $i,j\in I$, $a_i\in G_i$ and $a_j\in G_j$.
\end{itemize}
According to~\cite[Theorem~2.2]{RoRo87}, a~groupoid is differential if and only if it can be represented as an~$\mathbf{Lz}$-$\mathbf{Lz}$-sum satisfying the~left normal law. For more detailed information on differential groupoids, we refer to the~monograph~\cite[Secs.~5.6 and~8.4]{RoSm02}.

Let $n>0$. The~structure defined in~$\mathbf{Dm}$ by the~generators
$\{x,y\}$ and the~de\-fin\-ing relations $\{yx=y, xy^n=x\}$ is called the~{\it cycle\/} of length~$n$ and is denoted by~$\mathcal{D}_n$.
It is convenient to regard~$D_n$ as~$G_0\cup G_1$, where $G_1$ is the~singleton orbit $\{b\}$ and $G_0=\{a,ab,ab^2,\dots,ab^{n-1}\}$. We denote the~trivial groupoid by $\mathcal{D}_0$.

Let~$\mathbb{P}$ denote the~set of all primes; we assume that $\mathbb{P}=\{p_i\mid i<\omega\}$, where $p_i\leqslant p_j$ if and only if $i\leqslant j$
for all $i,j<\omega$. For an arbitrary set $F\in\mathcal{P}_\mathrm{fin}(\omega)$, we put
$[F]=\prod_{i\in F}p_i$ if $F\ne\varnothing$ and $[F]=1$ if $F=\varnothing$.

We will need the following basic properties of the cycles, see~\cite{Kr08,DG2} as well as~\cite{SZ2}.

\begin{lem}\label{d-1}
Let $n>0$.
\begin{enumerate}
\item
The~class $\{\mathcal{D}_m\mid m\ \text{divides}\ n\}$ coincides with the~class of nontrivial ho\-mo\-mor\-phic images of~$\mathcal{D}_n$.
\item
If~$m\in\omega$ and $\varphi\colon\mathcal{D}_n\to\mathcal{D}_m$ is a homomorphism then either $\varphi(\mathcal{D}_n)\cong\mathcal{D}_0$ or~$m$ divides~$n$
and $\varphi$ is onto.
\item
If $n>0$ and $X,X_1,\ldots,X_n\in\mathcal{P}_{\mathrm{fin}}(\omega)$ are such that the set $\{1,\ldots,n\}$ is minimal with respect to the property that $\mathcal{D}_{[X]}\in\mathbf{S}\mathbf{P}(\mathcal{D}_{[X_1]},\ldots,\mathcal{D}_{[X_n]})$, then $X=X_1\cup\ldots\cup X_n$.

\noindent
Conversely, if $X=X_1\cup\ldots\cup X_n$ then $\mathcal{D}_{[X]}\in\mathbf{S}\mathbf{P}(\mathcal{D}_{[X_1]},\ldots,\mathcal{D}_{[X_n]})$.
\end{enumerate}
\end{lem}

The structure of the~variety lattice of differential groupoids is explicitly described in~\cite{RoRo87}, see also~\cite[Theorem~8.4.14]{RoSm02}.
In particular, each proper subvariety of~$\mathbf{Dm}$ is defined by a~single identity within $\mathbf{Dm}$
and is locally finite.
In contrast to that description, the~structure of the~quasivariety lattice $\mathrm{Lq}(\mathbf{Dm})$ is much more complicated.
Namely, the~variety~$\mathbf{Dm}$ is $Q$-universal~\cite{Kr08}, there exist $2^\omega$ classes~$\mathbf{K}$ of differential groupoids such that
the~set of (isomorphism types of) finite sublattices of~$\mathrm{Lq}(\mathbf{K})$ is not computable~\cite{Sch15,SZ2}, and
there exist continuum many quasivarieties of differential groupoids with no independent quasi-equational basis~\cite{BNSZ}.

The~following assertions are proven in \cite{BNSZ,Kr08,DG2,DG}.

\begin{thm}\label{T:dg}
For each of the following properties, there exists continuum many quasivarieties of differential groupoids possessing this property{\rm :}
\begin{itemize}
\item Q-universality{\rm ;}
\item the undecidability of the set of {\rm (}isomorphism types{\rm )} of finite sublattices of the lattice of $\mathbf{K}$-subquasivarieties for a~suitable subclass $\mathbf{K}${\rm ;}
\item the existence of an~$\omega$-in\-de\-pen\-dent quasi-equational basis and the~lack of an independent quasi-equational basis within~$\mathbf{Dm}${\rm ;}
\item the existence of an independent quasi-equational basis{\rm ;}
\item the undecidability of the quasi-equational theory{\rm ;}
\item the undecidability of the finite membership problem and the~membership problem for finitely presented differ\-en\-tial groupoids.
\end{itemize}
\end{thm}

\subsection{Unary algebras}

As is proven in~\cite{Kr02}, the~variety~$\mathbf{K}_3$ of unary algebras of the type $\sigma=\{f,g\}$ defined by the~identities
\begin{align*}
& \forall x\,\forall y\ [f(f(x))=f(f(y))=f(g(y))],
\\
& \forall x\,\forall y\ [g(g(x))=g(g(y))=g(f(y))],
\\
& \forall x\ [f(f(x))=g(g(x))]
\end{align*}
is a~minimal $Q$-universal variety.
It follows from the~proof that the~proper subquasi\-va\-ri\-ety $\mathbf{V}\subseteq\mathbf{K}_3$ defined by the~quasi-identities
\begin{align*}
& \forall x\ [f(x)=f(f(x))\longrightarrow f(x)=g(x)],
\\
& \forall x\ [g(x)=g(g(x))\longrightarrow f(x)=g(x)],
\\
& \forall x\ [f(x)=g(x)\longrightarrow f(x)=f(f(x))],
\\
& \forall x\,\forall y\ [f(x)=f(y)\longrightarrow g(x)=g(y)],
\\
& \forall x\,\forall y\ [g(x)=g(y)\longrightarrow f(x)=f(y)]
\end{align*}
is $Q$-universal; moreover, so is the~lattice of $\mathbf{W}$-quasivarieties, where~$\mathbf{W}$ denotes the~subclass of~$\mathbf{V}$ defined by the~sentences
\begin{align}
& \forall x\,\forall y\ [g(x)=g(y)\,\&\,x\ne y\longrightarrow g(x)=g(g(x))], \label{16} \\
& \forall x\ [g(x)=g(g(x))]\longrightarrow \forall x\,\forall y\ [x=y]. \label{17}
\end{align}

We recall the~notation for certain unary algebras, see~\cite{Kr02,Kr16}.

For $n>1$, let~$\mathcal{C}_n$ denote the~algebra
whose universe is
$$
C_n=\{0\}\cup A_n\cup B_n\ \text{with}\ A_n=\{a_0^n,\dots,a_{n-1}^n\},\ B_n=\{b_0^n,\dots,b_{n-1}^n\}
$$
and the~operations are defined as follows:
$f(0)=g(0)=f(a_i^n)=g(a_i^n)=0$
and $g(b_i^n)=a_i^n$ for $0\leqslant i\leqslant n-1$,
$f(b_i^n)=a_{i+1}^n$ for $0\leqslant i\leqslant n-2$,
and $f(b_{n-1}^n)=a_0^n$.
Let $\mathcal{C}_1$ denote the~$2$-element algebra with the~universe $\{0,a\}$,
where $f(0)=g(0)=f(a)=g(a)=0$.
Let $\mathcal{C}_0$ denote the~trivial algebra.

It is clear that, for $n\geqslant 0$, we have~$\mathcal{C}_n\in\mathbf{W}$.

We recall necessary properties of the~algebras~$\mathcal{C}_n$
from~\cite{Kr02,Kr16,DG2}.

\begin{lem}\label{l-3}
If $n>1$ then the~following assertions hold.
\begin{enumerate}
\item
If~$m$ divides~$n$ then there exists a~homomorphism~$\varphi$ from~$\mathcal{C}_n$
onto~$\mathcal{C}_m${\rm ;} moreover, the~kernels of all such homomorphisms coincide and we have
$$
\ker\varphi=\{(x,x): x\in C_n\}\cup
\{(a_i,a_j), (b_i,b_j)\mid i\equiv j\pmod{m}\}.
$$
\item If $\mathcal{A}\in\mathbf{W}$ is nontrivial and there exists a~homomorphism from~$\mathcal{C}_n$ onto $\mathcal{A}$ then~$\mathcal{A}$ is isomorphic to~$\mathcal{C}_m$ for a~suitable divisor~$m$ of~$n$.
\item If $\mathcal{A}\in\mathbf{V}$ is nontrivial and there exists a~homomorphism $\varphi$ from $\mathcal{C}_n$ onto~$\mathcal{A}$ then one of the following conditions holds{\rm :}
\begin{enumerate}
\item $\mathcal{A}\in\mathbf{W}$ and assertion~{\rm (ii)} is valid{\rm ;}
\item the kernel $\ker\varphi$ contains a pair of one of the~forms $(0,a_i^n)$, $(0,b_i^n)$, $(a_i^n,b_j^n)$, $(a_i^n,a_j^n)$,
where the~difference $i-j$ is a~multiple of no divisor of~$n$, and the~structure~$\mathcal{A}$ satisfies the~premise of sentence~\eqref{17}{\rm ;}
\item the kernel $\ker\varphi$ contains a pair of the~form $(a_i^n,a_{i+k}^n)$ but $(b_i^n,b_{i+k}^n)\notin\ker\varphi$, where~$k$ is the~least positive number with this property, we have $k>1$, and~$k$ divides~$n$ {\rm (}in this case, the~algebra~$\mathcal{A}$
satisfies the~premise of sentence~\eqref{16} but violates its conclusion, i.e., $\mathcal{C}_k$ is a~substructure of~$\mathcal{A}$ and $|g^{-1}(\varphi(a_i^n))|>1${\rm )}.
\end{enumerate}
\end{enumerate}

If $X,Y,Z\in\mathcal{P}_\mathrm{fin}(\omega)$ and $n>1$
then the~following assertions hold.
\begin{enumerate}
\item[(iv)] There exists a~homomorphism from~$\mathcal{C}_{[X]}$ onto~$\mathcal{C}_{[Y]}$ if and only if $Y\subseteq X$.
\item[(v)] We have $\mathcal{C}_{[X]}\leq\mathcal{C}_{[Y]}\times\mathcal{C}_{[Z]}$ if and only if~$X=Y\cup Z$.
\end{enumerate}

If $n>1$ and $m>0$ then there exists a~quasi-identity~$q(n,m)$
such that, for every $k>1$, the~structure~$\mathcal{C}_{k}$ satisfies~$q(n,m)$ if and only if either~$k$ is not a~divisor of~$mn$ or~$k$ divides~$n$.
\end{lem}

The~following assertions are proven in \cite{Kr02,Kr16,DG2,DG}.

\begin{thm}\label{T:ua}
For each of the following properties, there exists continuum many $\mathbf{W}$-subquasivarieties possessing this property{\rm :}
\begin{itemize}
\item Q-universality{\rm ;}
\item the undecidability of the set of {\rm (}isomorphism types{\rm )} of finite sublattices of the lattice of $\mathbf{K}$-subquasivarieties for a~suitable subclass $\mathbf{K}${\rm ;}
\item the existence of an~$\omega$-in\-de\-pen\-dent quasi-equational basis and lack of an independent quasi-equational basis within~$\mathbf{V}${\rm ;}
\item the existence of an independent quasi-equational basis{\rm ;}
\item the undecidability of the quasi-equational theory{\rm ;}
\item the undecidability of the finite membership problem and the membership problem for finitely presented
unary algebras of the~type~$\{f,g\}$.
\end{itemize}
\end{thm}

\subsection{Standard and nonstandard quasivarieties}

For a structure $\mathcal{A}\in\mathbf{K}(\sigma)$, we say that $\mathbb{A}=\langle\mathcal{A},\tau\rangle$
is a {\it topological structure\/}
if $\tau$ is a topology on $A$ and all the basic operations of $\mathcal{A}$ are continuous and all the basic relations on $\mathcal{A}$
are closed with respect to $\tau$. For a topological structure $\mathbb{A}$, we denote its algebraic reduct by $\mathcal{A}$ and its topology by $\tau_\mathbb{A}$.
A topology $\tau$ on a set $A$ is {\it Boolean} if the topological space $\langle A,\tau\rangle$ is compact Hausdorff and has a base of clopen sets.
A topological structure $\mathbb{A}$ is {\it Boolean} if $\tau_\mathbb{A}$ is Boolean.
For a class $\mathbf{K}$ of topological structures of a fixed type,
let $\mathbf{S}_c(\mathbf{K})$ denote the class of structures that are isomorphic to closed substructures of structures from $\mathbf{K}$.

A~structure is {\it profinite\/} ({\it with respect to\/} $\mathbf{K}$) if it is isomorphic to the inverse limit of a~family of finite structures
(from~$\mathbf{K}$).
Each profinite structure is naturally equipped with a~Boolean topology (in this case, this is the~product topology).
A prevariety $\mathbf{K}$ is {\it profinite} if each Boolean topological structure with its algebraic reduct belonging to $\mathbf{K}$ is profinite with respect to $\mathbf{K}$.
A universal Horn class $\mathbf{K}$ is {\it standard} if each Boolean topological structure with its algebraic reduct in $\mathbf{K}$
belongs to $\mathbf{S}_c\mathbf{P}^+(\mathbf{K}_{\rm{fin}})$, where all the finite structures are equipped with
the discrete topology and the~Cartesian product is equipped with the product topology.

It is immediate from~\cite[Theorem 2.6 and Lemma 1.4(ii)]{DCJ} that a universal Horn class $\mathbf{K}$ is standard if and only if it is profinite.
  
%The following statement is almost straightforward.}

%\begin{lem}
%red{A universal Horn class $\mathbf{K}$ is standard if and only if it is profinite.}
%\end{lem}

%\begin{proof}
%\red{It follows from \cite[Theorem 2.6]{DCJ} that $\mathbf{K}$ is profinite whenever $\mathbf{K}$ is standard.
%Conversely, suppose that $\mathbf{K}$ is profinite and let $\mathbb{A}$ be a Boolean topological structure such that $\mathcal{A}\in\mathbf{K}$.
%Then $\mathbb{A}$ is profinite with respect to $\mathbf{K}$; that is, there is a nonempty set $I$ and finite structures $\mathcal{A}_i\in\mathbf{K}$ such that
%$\mathbb{A}\cong\varprojlim_{i\in I}\mathbb{A}_i$, where $\tau_{\mathbb{A}_i}$ is discrete for all $i\in I$.
%By \cite[Lemma 1.4(ii)]{DCJ}, $\mathbb{A}\in\mathbf{S}_c\mathbf{P}^+(\mathbf{K}_{\rm{fin}})$.}
%\end{proof}

For more information and results on natural dualities, topological quasiva\-ri\-eties,
Boolean topological structures, and topology, the reader is referred to~\cite{CD,DCJ,J}.

We will need the following property of the inverse limit, see~\cite[Lemma 3.2]{DCJ}.

\begin{lem}\label{3.2}
Let $\sigma$ contain no relation symbols,
let $\mathcal{A}=\varprojlim_{i\in I}\mathcal{A}_i$, where $\Lambda$ is a surjective inverse spectrum and $\mathcal{A}_i$ is a finite structure for every $i\in I$,
let $\mathcal{B}$ be a finite structure, and let $\varphi\colon\mathcal{A}\to\mathcal{B}$ be a {\rm (}continuous{\rm )} homomorphism.
Then there exist $i\in I$ and a {\rm (}continuous{\rm )} homomorphism $\psi\colon\mathcal{A}_i\to\mathcal{B}$ such that $\varphi=\psi\pi_i$.
\end{lem}

The following notion from~\cite{DCJ} and its connection with standard quasivarieties is an essential tool in the proof of our main results.

\begin{definition}
Let $\mathbf{M}\subseteq\mathbf{K}(\sigma)$, where $\sigma$ contains no relation symbols.
A structure $\mathcal{A}$ is {\it pointwise non-separable\/} with respect to $\mathbf{M}$ if
there exist $a_1,a_2\in A$ such that $a_1\ne a_2$ and $\varphi(a_1)=\varphi(a_2)$
for every finite structure $\mathcal{B}\in\mathbf{M}$ and every homomorphism $\varphi\colon\mathcal{A}\to\mathcal{B}$.
\end{definition}

The following assertion is immediate from~\cite[Lemmas~3.3 and~3.4]{DCJ}.

\begin{lem}\label{DC}
Let $\mathcal{A}=\varprojlim_{i\in I}\mathcal{A}_i$, where $\Lambda$ is a surjective inverse spectrum and $\mathcal{A}_i$ is a finite structure for every $i\in I$,
and let $\mathbf{K}$ be a quasivariety.
If $\mathcal{A}$ is pointwise non-separable with respect to $\mathbf{K}$
then $\mathcal{A}$ is not profinite with respect to $\mathbf{K}$.

In particular, if $\mathcal{A}\in\mathbf{K}$ then $\mathbf{K}$ is not standard.
\end{lem}

\section{Main results}

We formulate and prove similar results for $\mathbf{Dm}$ and~$\mathbf{V}$.
Namely, we find

(a) continuum many nonstandard subquasivarieties with no independent quasi-equational bases;

(b) and continuum many nonstandard subquasivarieties having an~independent quasi-equational basis.


\subsection{Differential groupoids}

\begin{thm}\label{T:1dg}
There exist continuum many subquasivarieties of $\mathbf{Dm}$ that are not standard and have no independent quasi-equational basis.
\end{thm}

\begin{proof}
Let $I\subseteq\omega$ and let $F\in\mathcal{P}_{\mathrm{fin}}(\omega)$.
We denote by $\varphi_F^I$ the~quasi-identity
\[
\forall x\,\forall y\  xy^{[F]}=x\ \&\ yx=y\ \to\ xy^{[F\cap I]}=x.
\]
Let $\Phi_I=\{\varphi_F^I\mid F\in \mathcal{P}_{\mathrm{fin}}(\omega)\}$
and let $\mathbf{K}_I$ be the~subquasivariety of $\mathbf{Dm}$ defined by the set of quasi-identities~$\Phi_I$.

Let $I\in\mathcal{P}_{\mathrm{inf}}(\omega)$, $F\in\mathcal{P}_{\mathrm{fin}}(\omega)$, and $p\in\omega{\setminus}F$.
We denote by~$\psi_F^p$ the~quasi-identity
\[
\forall x\,\forall y\  xy^{[F\cup\{p\}]}=x\ \&\ yx=y\ \to\ xy^{[F]}=x.
\]
We put $\Psi_p=\{\psi_F^p\mid F\in P_{\rm fin}(\omega),\ p\in\omega{\setminus}F\}$ and $\Psi_I=\bigcup_{p\in\omega{\setminus}I}\Psi_p$.
We denote by $\mathbf{K}^\prime_I$ the~subquasivariety of~$\mathbf{Dm}$ defined by the set of quasi-identities~$\Psi_I$.

The following assertion is proven in~\cite[Theorem~4]{DG}.

\begin{prop}\label{P:dg}
We have $\mathbf{K}_I=\mathbf{K}^\prime_I$ for every $I\in\mathcal{P}_{\mathrm{inf}}(\omega)$. The set $\Psi_I$ is an~$\omega$-independent quasi-equational basis of~$\mathbf{K}$. There is no independent quasi-equational basis of~$\mathbf{K}$.
\end{prop}

We fix $I\in\mathcal{P}_{\mathrm{inf}}(\omega)$ and $k\in\omega{\setminus}I$.
Assume that the numbers in~$I$ are ordered in the natural way, i.e., we have $I=\{i_n\mid n<\omega\}$, where $i_n\leqslant i_m$ if and only if $n\leqslant m$ for all $m,n<\omega$.
For every $n<\omega$, we put
\[
F_n=\{i_0,\ldots,i_n\},\enskip
\mathcal{A}=\mathcal{D}_{p_k},\enskip
\mathcal{A}_n=\mathcal{D}_{[F_n]},\enskip
\mathcal{B}_n=\mathcal{D}_{[\{k\}\cup F_n]}.
\]
By the~definition of~$\mathbf{K}_I$, we immediately obtain the following assertion.

\setcounter{claim}{0}
\begin{claim}\label{C:1}
For every $F\in\mathcal{P}_{\mathrm{fin}}(\omega)$, we have $\mathcal{D}_{[F]}\in\mathbf{K}_I$ if and only if $F\subseteq I$.
\end{claim}

In particular, we have $\mathcal{A}\notin\mathbf{K}_I$ and $\mathcal{A}_n\in\mathbf{K}_I$ but $\mathcal{B}_n\notin\mathbf{K}_I$ for every $n<\omega$.

We construct an inverse spectrum $\Lambda$.
By Lemma~\ref{d-1}, if $i<j<\omega$ then there is a surjective homomorphism
$\pi_{ij}\colon\mathcal{B}_j\to\mathcal{B}_i$; moreover, we may assume that $\pi_{ij}(b)=b$ and $\pi_{ij}(a)=a$,
where $\{b\}$ is the singleton orbit and $a$ is the second generator of~$\mathcal{D}_n$ for every $n<\omega$.
According to the definition of the inverse limit, we denote by $\pi_{ii}$ the identity automorphism on~$\mathcal{B}_i$
for every $i<\omega$.

We immediately obtain the following assertion.

\begin{claim}\label{C:5}
The triple $\Lambda=\langle\omega,\mathcal{B}_j,\pi_{ij}\rangle$ is a surjective inverse spectrum.
\end{claim}

We put $\mathcal{B}=\varprojlim\Lambda$.
Since each $\pi_{ij}$ fixes~$a$ and~$b$, we conclude that $B\ne\varnothing$.
Moreover, we have $\mathcal{B}\in\mathbf{S}\mathbf{P}(\mathbf{Dm})\subseteq\mathbf{Dm}$.
For every $n<\omega$, we denote by $\pi_n$ the restriction of the $n$th projection $\prod_{n<\omega}B_n\to B_n$ to~$B$.

\begin{claim}\label{C:2}
The differential groupoid~$\mathcal{B}$ is infinite and belongs to~$\mathbf{K}_I$.
\end{claim}

\begin{scproof}
We show first that $\mathcal{B}\models\Phi_I$.
Indeed, consider an arbitrary quasi-identity $\varphi_F^I\in\Phi_I$.
Suppose that the premise of~$\varphi_F^I$ holds in~$\mathcal{B}$ under an interpretation $\gamma\colon \{x,y\}\to B$.
Then there exists a homomorphism $f\colon\mathcal{D}_{[F]}\to\mathcal{B}$ such that $f(a)=\gamma(x)$ and $f(b)=\gamma(y)$.
For all $n<\omega$ and $u\in D_{[F]}$, we put $f_n(u)=\pi_n(f(u))$.
We find that $f_n$ is a homomorphism from $\mathcal{D}_{[F]}$ to $\mathcal{B}_n$.
By Lemma~\ref{d-1}, we have either $f_n(\mathcal{D}_{[F]})\cong\mathcal{D}_{[G]}\leq\mathcal{B}_n$ for some nonempty $G\subseteq F$
or $f_n(\mathcal{D}_{[F]})\leq\mathcal{D}_1$.
Since all proper subgroupoids of $\mathcal{B}_n$ are left-zero modes, we conclude that either
$f_n(\mathcal{D}_{[F]})\cong\mathcal{B}_n$, or $f_n(\mathcal{D}_{[F]})\cong\mathcal{D}_1$, or $f_n(\mathcal{D}_{[F]})\cong\mathcal{D}_0$.
In the first case, we have $|B_n|\leqslant [F]+1$.
Since $\mathcal{D}_{[F]}$ is a finite groupoid and $|B_n| <|B_{n+1}|$ for each $n<\omega$,
there is $s<\omega$ such that $[F]+1<|B_n|$ for all $n\geqslant s$.
This implies that $f_n(\mathcal{D}_{[F]})$ is isomorphic to a subgroupoid of $\mathcal{D}_1$ if $n\geqslant s$.
We denote $J_s=\{n<\omega\mid n\geqslant s\}$.

We summarize the above conclusions as follows.
The differential groupoid $\mathcal{B}$ is isomorphic to the~inverse limit $\varprojlim\langle J_s,\mathcal{B}_j,\pi_{ij}\rangle$
and is a~subdirect product of the~family of differential groupoids $\{\mathcal{B}_n\mid n\in J_s\}$.
The differential groupoid $f(\mathcal{D}_{[F]})$ is a subdirect product of the~family $\{f_n(\mathcal{D}_{[F]})\mid n\in J_s\}$
of subgroupoids of $\mathcal{D}_1$; hence, it is a left-zero mode.

We conclude that the subgroupoid $f(\mathcal{D}_{[F]})$ of $\mathcal{B}$ generated by the set $\{a,b\}$ is a left-zero mode.
But then the conclusion of $\varphi_F^I$ holds in~$\mathcal{B}$.

Finally, if the universe~$B$ were finite then, by Lemma~\ref{3.2}, we would obtain $|B_{n+1}|\leqslant|B|\leqslant|B_n|$ for some $n<\omega$.
Since $|B_{n+1}|>|B_n|$, we arrive at a contradiction.
\end{scproof}

We remind that $\mathcal{A}\notin\mathbf{K}_I$,
$\mathcal{A}_n\in\mathbf{K}_I$, and $\mathcal{B}_n\notin\mathbf{K}_I$ for every $n<\omega$.
By Lemma~\ref{d-1}, the differential groupoid $\mathcal{B}_n$ is a subdirect product of the differential groupoids $\mathcal{A}$ and $\mathcal{A}_n$.
We denote the corresponding projections (which are onto homomorphisms) by
$\alpha_n\colon\mathcal{B}_n\to\mathcal{A}$ and $\beta_n\colon\mathcal{B}_n\to\mathcal{A}_n$.
As $\mathcal{B}_n\notin\mathbf{K}_I$ and $\mathcal{A}_n\in\mathbf{K}_I$,
the kernel $\ker\beta_n$ is the least congruence on $\mathcal{B}_n$ for no $n<\omega$.
For each $n<\omega$, we can find distinct elements $b^n_1,b^n_2\in B_n$ such that $\beta_n(b^n_1)=\beta_n(b^n_2)$.
We conclude that $\alpha_n(b^n_1)\ne\alpha_n(b^n_2)$.

By Lemma~\ref{d-1}, if $i<j<\omega$ then there is a surjective homomorphism
$\varphi_{ij}\colon\mathcal{A}_j\to\mathcal{A}_i$; moreover, we may assume that $\varphi_{ij}(b)=b$ and $\varphi_{ij}(a)=a$,
where $\{b\}$ is the singleton orbit and $a$ is the second generator of~$\mathcal{D}_n$ for every $n<\omega$.
It is easy to see that, for all $i,j<\omega$ with $i<j$ and all $u\in B_j$, we have $\beta_i(\pi_{ij}(u))=\varphi_{ij}(\beta_j(u))$.

The proof of the following assertion repeats the proof of Claim~4 from the proof of \cite[Theorem 4]{KrNuSchw21}.

\begin{claim}\label{C:3}
If $(b_1,b_2)\in\ker\beta_j\setminus\ker\alpha_j$ for some $j<\omega$ then, for $i\leqslant j$, we have
\[
\bigl(\pi_{ij}(b_1),\pi_{ij}(b_2)\bigr)\in\ker\beta_i\setminus\ker\alpha_i.
\]
\end{claim}

\begin{claim}\label{C:4}
The differential groupoid $\mathcal{B}$ is pointwise non-separable with respect to~$\mathbf{K}_I$.
\end{claim}

\begin{scproof}
We consider the set
\[
H_{-1}=\bigcup_{n<\omega}\bigl\{(h^n_1,h^n_2)\in B^2_n\mid(h^n_1,h^n_2)\in\ker\beta_n\setminus\ker\alpha_n\bigr\}.
\]
Since $(b^n_1,b^n_2)\in H_{-1}\cap B^2_n\ne\varnothing$ for every $n<\omega$, the set $H_{-1}$ is infinite.
Since the set $B_0$ is finite, there is a pair $\overline{c}_0$ of elements of $B_0$ such that the set
\[
H_0=\bigcup_{n<\omega}\bigl\{(h^n_1,h^n_2)\in H_{-1}\mid\bigl(\pi_{0n}(h^n_1),\pi_{0n}(h^n_2)\bigr)=\overline{c}_0\bigr\}
\]
is infinite. By Claim~\ref{C:3}, we have $\overline{c}_0\in\ker\beta_0\setminus\ker\alpha_0$.

We use induction on~$i$ and construct a chain of infinite sets $H_i$, $i<\omega$.
Assume that, for some $i<\omega$, we have already found pairs
$\overline{c}_j\in\ker\beta_j{\setminus}\ker\alpha_j$ for $0\leqslant j\leqslant i$ such that the set
\[
H_j=\bigcup_{n\geqslant j}\bigl\{(h^n_1,h^n_2)\in H_{j-1}\mid\bigl(\pi_{jn}(h^n_1),\pi_{jn}(h^n_2)\bigr)=\overline{c}_j\bigr\}
\]
is infinite and $\pi_{jj^\prime}$ takes the pair $\overline{c}_{j^\prime}$ into the pair $\overline{c}_j$
whenever $0\leqslant j\leqslant j^\prime\leqslant i$.
Since $H_i$ is an infinite set and $B_i$ and $B_{i+1}$ are finite sets,
there is a pair $\overline{c}_{i+1}$ of elements of $B^m_{i+1}$ such that the set
\[
H_{i+1}=\bigcup_{n\geqslant i+1}\bigl\{(h^n_1,h^n_2)\in H_i\mid\bigl(\pi_{i+1,n}(h^n_1),\pi_{i+1,n}(h^n_2)\bigr)=\overline{c}_{i+1}\bigr\}
\]
is infinite. By Claim~\ref{C:3}, we have $\overline{c}_{i+1}\in\ker\beta_{i+1}\setminus\ker\alpha_{i+1}$.
Moreover, $\overline{c}_{i+1}\in H_{i+1}\subseteq H_i$.
We conclude therefore that $\pi_{i,i+1}$ takes the pair $\overline{c}_{i+1}$ into the pair $\overline{c}_i$.
By the definition of an inverse spectrum and the induction hypothesis, we deduce that
$\pi_{j,i+1}(\overline{c}_{i+1})=\pi_{ji}\bigl(\pi_{i,i+1}(\overline{c}_{i+1})\bigr)=\pi_{ji}(\overline{c}_i)=\overline{c}_j$ whenever $j<i+1$.

We introduce a pair $\overline{c}=(c_1,c_2)$ of elements of~$B$ such that
for all $n<\omega$ and $i\in\{1,2\}$, $\pi_i(c_i)$ is the $i$th component of $\overline{c}_n$.

We consider a homomorphism $f\colon\mathcal{B}\to\mathcal{D}$, where $\mathcal{D}\in\mathbf{K}_I$ and $\mathcal{D}$ is a finite differential groupoid.
By Lemma~\ref{3.2}, there exist $n<\omega$ and a homomorphism $g\colon\mathcal{B}_n\to\mathcal{D}$ such that
$f=g\pi_n$.
By Lemma~\ref{d-1} and Claim~\ref{C:1}, we obtain $g(\mathcal{B}_n)\cong\mathcal{D}_{[G]}\in\mathbf{K}_I$
for some $G\subseteq(\{k\}\cup F_n)\cap I=F_n$ or $g(\mathcal{B}_n)\cong\mathcal{D}_0$.
We can find therefore a~homomorphism $h\colon\mathcal{A}_n\to\mathcal{D}$ with $g=h\beta_n$.
We conclude that $f=g\pi_n=h\beta_n\pi_n$ whence $\ker\beta_n\pi_n\subseteq\ker f$.
As $\overline{c}_n\in\ker\beta_n$, we obtain $(c_1,c_2)\in\ker\beta_n\pi_n\subseteq\ker f$.
\end{scproof}

By Claims~\ref{C:2} and~\ref{C:4} and Lemma~\ref{DC}, the quasivariety~$\mathbf{K}_I$ is not standard.

It remains to notice that, in view of Claim~\ref{C:1}, the quasivarieties of the form~$\mathbf{K}_I$
are pairwise distinct.
\end{proof}

Following~\cite{GoKr01}, a set $\Phi$ of first-order sentences is said to be {\it directed\/} if,
for arbitrary $\varphi_0,\varphi_1\in\Phi$, there is $\varphi\in\Phi$ such that both $\varphi_0$ and $\varphi_1$
are consequences of $\varphi$.
Directed sets of first-order sentences are, in a sense, antagonists of independent sets of sentences.

\begin{thm}\label{T:2dg}
There exist continuum many subquasivarieties in $\mathbf{Dm}$ that are not standard and have an independent quasi-equational basis
$[$a directed quasi-equational basis, respectively$]$.
\end{thm}

\begin{proof}
This proof is similar to the one of Theorem \ref{T:1dg} but less complicated.

We consider an infinite set $I=\{i_n\mid n<\omega\}$ and assume that its members are ordered in the natural way. That is,
we have
$i_n\leqslant i_m$ if and only if $n\leqslant m$ for all $m,n<\omega$.
We put $F_{-1}=\varnothing$, $F_n=\{i_0,\ldots,i_n\}$, and $\mathcal{B}_n=\mathcal{D}_{[F_n]}$ for every $n<\omega$.

For each $m<\omega$, let $\xi_m$ denote the quasi-identity
\[
\forall x\,\forall y\  xy^{[F_m]}=x\ \&\ yx=y\ \to\ xy^{[F_{m-1}]}=x.
\]
We denote
\[
\Xi_I=\{\xi_m\mid m<\omega\}\quad
\text{and}\quad\mathbf{M}_I=\operatorname{Mod}\Xi_I\cap\mathbf{Dm}.
\]

\setcounter{claim}{0}
\begin{claim}\label{C:21}
The set~$\Xi_I$ is an independent quasi-equational basis of $\mathbf{M}_I$ within~$\mathbf{Dm}$.
The quasivariety $\mathbf{M}_I$ consists of all structures $\mathcal{A}\in\mathbf{Dm}$ with the following property{\rm :}
\[
\text{if~} F\in\mathcal{P}_{\mathrm{fin}}(\omega) \text{~is nonempty and~} \mathcal{D}_{[F]}\in\mathbf{S}(\mathcal{A}) \text{~then~} F\nsubseteq I.
\]
\end{claim}

\begin{scproof}
It is not difficult to see that $\mathcal{B}_n$ satisfies $\xi_m$ if $m\ne n$ and violates $\xi_n$.
Hence, $\Xi_I$ is an independent set of quasi-identities.

If $\mathcal{D}_{[F]}$ is embeddable into $\mathcal{A}$ for no $F\in\mathcal{P}_{\mathrm{fin}}(\omega)$ with $\varnothing\ne F\subseteq I$
then, for every $m<\omega$, the premise of the quasi-identity $\xi_m$ can hold under no interpretation
$\gamma\colon\{x,y\}\to A$ in view of Lemma~\ref{d-1}.
Therefore, $\mathcal{A}\models\xi_m$ for all $m<\omega$ and $\mathcal{A}\in\mathbf{M}_I$.
Conversely, let $\mathcal{A}\in\mathbf{M}_I$ and let $\mathcal{D}_{[F]}$ embed into $\mathcal{A}$ for some nonempty set
$F\in\mathcal{P}_{\mathrm{fin}}(\omega)$ with $F\subseteq I$. Let $i_n$ be the greatest element of $F$; then
$F\subseteq F_n$.
Therefore, the differential groupoid $\mathcal{D}_{[F]}$ satisfies the premise of $\xi_n$
under the interpretation $\gamma$ with $\gamma(x)=a$, $\gamma(y)=b$ but violates the conclusion of $\xi_n$ under the same interpretation $\gamma$
in view of Lemma~\ref{d-1}. Hence, $\mathcal{A}\not\models\xi_n$ which contradicts our assumption $\mathcal{A}\in\mathbf{M}_I$.
\end{scproof}

Notice that Claim~\ref{C:21} also provides us with a directed quasi-equational basis of $\mathbf{M}_I$ within~$\mathbf{Dm}$.
Namely, for each nonempty set $F\in\mathcal{P}_{\mathrm{fin}}(\omega)$ with $F\subseteq I$, we introduce the following quasi-identity~$\chi_F$:
\[
\forall x\,\forall y\  xy^{[F]}=x\ \&\ yx=y\ \to\ xy=x.
\]
We consider the set $X=\{\chi_F\mid F\in\mathcal{P}_{\mathrm{fin}}(\omega)\text{~and~} \varnothing\ne F\subseteq I\}$.

\begin{claim}\label{C:dir}
The~set $X$ forms a directed quasi-equational basis of $\mathbf{M}_I$ within~$\mathbf{Dm}$.
\end{claim}

\begin{scproof}
It is clear that $X$ is a quasi-equational basis of $\mathbf{M}_I$ within~$\mathbf{Dm}$.
Moreover, the quasi-identities $\chi_F,\chi_G\in X$
are consequences of the quasi-identity $\chi_{F\cup G}$ for all nonempty sets $F,G\in\mathcal{P}_{\mathrm{fin}}(\omega)$.
\end{scproof}

We construct an inverse spectrum $\Lambda=\langle\omega,\mathcal{B}_j,\pi_{ij}\rangle$ as follows.
According to Lemma~\ref{d-1}, for all $i,j<\omega$ with $i<j$, there is a surjective homomorphism
$\pi_{ij}\colon\mathcal{B}_j\to\mathcal{B}_i$; moreover, we may assume that $\pi_{ij}(b)=b$ and $\pi_{ij}(a)=a$,
where $\{b\}$ is the singleton orbit and $a$ is the second generator of~$\mathcal{D}_n$ for each $n<\omega$.
According to the definition of an inverse spectrum, let $\pi_{ii}$ be the identity automorphism of~$\mathcal{B}_i$
for each $i<\omega$.

We immediately obtain the following assertion.

\begin{claim}\label{C:22}
The triple $\Lambda=\langle\omega,\mathcal{B}_j,\pi_{ij}\rangle$ is a surjective inverse spectrum.
\end{claim}

We put $\mathcal{B}=\varprojlim\Lambda$.
Since each $\pi_{ij}$ fixes~$a$ and~$b$, we conclude that $B\ne\varnothing$.
Moreover, we have $\mathcal{B}\in\mathbf{S}\mathbf{P}(\mathbf{Dm})\subseteq\mathbf{Dm}$.
For every $n<\omega$, we denote by $\pi_n$ the restriction of the $n$th projection $\prod_{n<\omega}B_n\to B_n$ to~$B$.

\begin{claim}\label{C:23}
The differential groupoid~$\mathcal{B}$ is infinite and belongs to~$\mathbf{M}_I$.
\end{claim}

\begin{scproof}
The same arguments as in the proof of Theorem \ref{T:1dg} (see Claim~\ref{C:2} there) show that $\mathcal{B}$ is infinite.

We prove that $\mathcal{B}\in\mathbf{M}_I$. In view of Claim \ref{C:dir}, it suffices to prove that $\mathcal{B}\models X$.
Assume that $F\in\mathcal{P}_{\mathrm{fin}}(\omega)$ and $\varnothing\ne F\subseteq I$.
If the premise of $\chi_F$ holds in~$\mathcal{B}$ under an interpretation $\gamma\colon\{x,y\}\to B$
then there exists a homomorphism $f\colon\mathcal{D}_{[F]}\to\mathcal{B}$ such that $f(a)=\gamma(x)$ and $f(b)=\gamma(y)$.

For all $n<\omega$ and $u\in D_{[F]}$, we put $f_n(u)=\pi_nf(u)$.
We find that $f_n$ is a homomorphism from $\mathcal{D}_{[F]}$ to $\mathcal{B}_n$.
By Lemma~\ref{d-1}, we have either $f_n(\mathcal{D}_{[F]})\cong\mathcal{D}_{[G]}\leq\mathcal{B}_n$ for some $G\subseteq F$
or $f_n(\mathcal{D}_{[F]})\cong\mathcal{D}_0$.
This means that $f_n(\mathcal{D}_{[F]})$ is either a subgroupoid of $\mathcal{D}_1$ or isomorphic to $\mathcal{B}_n$.
In the second case, we have $|B_n|\leqslant [F]+1$.
Since $\mathcal{D}_{[F]}$ is a finite structure and $|B_n| <|B_{n+1}|$ for each $n<\omega$,
there is $s<\omega$ such that $[F]+1<|B_n|$ whenever $n\geqslant s$.
Therefore, the differential groupoid $f_n(\mathcal{D}_{[F]})$ with $n\geqslant s$ is a subgroupoid of $\mathcal{D}_1$.

The same arguments as in the proof of Theorem~\ref{T:1dg} (see Claim~\ref{C:2} there) show that the subgroupoid
$f(\mathcal{D}_{[F]})$ of $\mathcal{B}$ generated by the set $\{a,b\}$ is a left-zero mode,
i.e., the conclusion of $\chi_F$ holds in~$\mathcal{B}$.
\end{scproof}

\begin{claim}\label{C:24}
The differential groupoid $\mathcal{B}$ is pointwise non-separable with respect to~$\mathbf{M}_I$.
\end{claim}

\begin{scproof}
As $\pi_n$ maps $\mathcal{B}$ onto $\mathcal{B}_n$ for all $n<\omega$, we have $\mathcal{B}\notin\mathbf{Q}(\mathcal{D}_1)$.

Consider an arbitrary homomorphism $f\colon\mathcal{B}\to\mathcal{D}$, where $\mathcal{D}\in\mathbf{M}_I$ is a finite differential groupoid.
By Lemma \ref{3.2}, there are $n<\omega$ and $g\colon\mathcal{B}_n\to\mathcal{D}$ such that $f=g\pi_n$ and $g$ is a~homomorphism.
By Lemma \ref{d-1}, we have either $g(\mathcal{B}_n)\cong\mathcal{D}_{[G]}\in\mathbf{M}_I$ for some $G\subseteq F_n\subseteq I$
or $g(\mathcal{B}_n)\cong\mathcal{D}_0$.
Since $\mathcal{D}_{[G]}\in\mathbf{M}_I$, we have $G=\varnothing$ by Claim~\ref{C:dir} in the first case.
This means that $f(\mathcal{B})\cong g(\mathcal{B}_n)\leq\mathcal{D}_1$ in any case,
which contradicts the fact that $\mathcal{B}\notin\mathbf{Q}(\mathcal{D}_1)$.
\end{scproof}

By Claims \ref{C:23} and \ref{C:24} and Lemma~\ref{DC}, the quasivariety $\mathbf{M}_I$ is not standard.

It remains to notice that, in view of Claim~\ref{C:21}, the quasivarieties of the form~$\mathbf{M}_I$
are pairwise distinct.
\end{proof}

\subsection{Unary algebras}

\begin{thm}\label{T:1ua}
There exist continuum many subquasivarieties of $\mathbf{V}$ that are not standard
and have no independent quasi-equational basis.
\end{thm}

\begin{proof}
Let $I\subseteq\omega$ and let $F\in\mathcal{P}_{\mathrm{fin}}(\omega)$.
We denote by $\varphi_F^I$ the~quasi-identity
\begin{multline*}
\forall x\,\forall y_0\dots\forall y_{[F]-1}\,\forall z_0\dots\forall z_{[F]-1}\,
\bigl[f(x)=x\,\wedge\, g(x)=x\,\wedge
\\
\wedge \bigwedge_{0\leqslant i<[F]} f(y_i)=x\,\wedge\,\bigwedge_{0\leqslant i<[F]} g(y_i)=x\,
\wedge\,\bigwedge_{0\leqslant i<[F]} g(z_i)=y_i\,\wedge
\\
\wedge \bigwedge_{0\leqslant i<[F]-1} f(z_i)=y_{i+1}\,\wedge\,f(z_{[F]-1})=y_0\bigr]
\longrightarrow
g(z_0)=y_{[F\cap I]}.
\end{multline*}
This is the~quasi-identity $q(n,m)$ from Lemma~\ref{l-3} with $n=[F\cap I]$, $m=[F\setminus I]$.

Let $\Phi_I=\{\varphi_F^I\mid \varnothing\ne F\in\mathcal{P}_{\mathrm{fin}}(\omega)\}$
and let $\mathbf{K}_I$ be the~subquasivariety of $\mathbf{V}$ defined by the set of quasi-identities~$\Phi_I$.

The following assertion is proven in~\cite[Theorem~8]{DG}.

\begin{prop}\label{P:ua}
Each of the quasivarieties $\mathbf{K}_I$ admits an~$\omega$-independent quasi-equational basis  but
lacks an independent quasi-equational basis.
\end{prop}

We fix $I\in\mathcal{P}_{\mathrm{inf}}(\omega)$ and $k\in\omega{\setminus}I$.
Assume again that $I=\{i_n\mid n<\omega\}$ and $i_n\leqslant i_m$ if and only if $n\leqslant m$ for all $m,n<\omega$.
For every $n<\omega$, we put
\[
F_n=\{i_0,\ldots,i_n\},\enskip
\mathcal{A}=\mathcal{C}_{p_k},\enskip
\mathcal{A}_n=\mathcal{C}_{[F_n]},\enskip
\mathcal{B}_n=\mathcal{C}_{[\{k\}\cup F_n]}=\mathcal{C}_{p_k\cdot[F_n]}.
\]
By the~definition of~$\mathbf{K}_I$ and Lemma~\ref{l-3}, we immediately obtain the following assertion.

\setcounter{claim}{0}
\begin{claim}\label{C:1u}
For every $F\in\mathcal{P}_{\mathrm{fin}}(\omega)$, we have $\mathcal{C}_{[F]}\in\mathbf{K}_I$ if and only if $F\subseteq I$.
\end{claim}

In particular, we have $\mathcal{A}\notin\mathbf{K}_I$ and $\mathcal{A}_n\in\mathbf{K}_I$ but $\mathcal{B}_n\notin\mathbf{K}_I$ for every $n<\omega$.

We construct an inverse spectrum $\Lambda$.
By Lemma~\ref{l-3}, if $i<j<\omega$ then there is a surjective homomorphism
$\pi_{ij}\colon\mathcal{B}_j\to\mathcal{B}_i$; moreover, we may assume that $\pi_{ij}(0)=0$,
$\pi_{ij}(a^p_u)=a^q_v$, and $\pi_{ij}(b^p_u)=b^q_v$, where $p=p_k\cdot[F_j]$, $q=p_k\cdot[F_i]$, and $u\equiv v\pmod{q}$.
According to the definition of an inverse spectrum, let $\pi_{ii}$ be the identity automorphism on~$\mathcal{B}_i$
for every $i<\omega$.

The following assertion holds.

\begin{claim}\label{C:5u}
The triple $\Lambda=\langle\omega,\mathcal{B}_j,\pi_{ij}\rangle$ is a surjective inverse spectrum.
\end{claim}

We put $\mathcal{B}=\varprojlim\Lambda$.
Since each homomorphism $\varphi_{ij}$ fixes the element~$0$, we conclude that $B\ne\varnothing$.
Moreover, we have $\mathcal{B}\in\mathbf{S}\mathbf{P}(\mathbf{V})\subseteq\mathbf{V}$.
For each $n<\omega$, we denote by $\pi_n$ the restriction of the $n$th projection $\prod_{n<\omega}\mathcal{B}_n\to\mathcal{B}_n$ to~$\mathcal{B}$.
As there $\pi_n$ is onto for all $n<\omega$, the following statement is true.

\begin{claim}\label{C:g}
We have $\mathcal{B}\not\models\forall x\,g(g(x))=g(x)$.
\end{claim}

\begin{claim}\label{C:2u}
The algebra~$\mathcal{B}$ is infinite and belongs to~$\mathbf{K}_I$.
\end{claim}

\begin{scproof}
The same arguments as in the proof of Theorem~\ref{T:1dg} (see the proof of Claim~\ref{C:2} there) show that $\mathcal{B}$ is infinite.
We prove that $\mathcal{B}\models\Phi_I$.

We consider a quasi-identity $\varphi_F^I\in\Phi_I$ with $F\ne\varnothing$ and assume that
the premise of~$\varphi_F^I$ holds in~$\mathcal{B}$ under an interpretation
$\gamma\colon\{x,y_0,\dots,y_{[F]-1},z_0,\dots,z_{[F]-1}\}\to B$.
Then there exists a homomorphism $\varphi\colon\mathcal{C}_{[F]}\to\mathcal{B}$
such that $\varphi(0)=\gamma(x)$, $\varphi(a_i^{[F]})=\gamma(y_i)$, and $\varphi(b_i^{[F]})=\gamma(z_i)$ for $i<[F]$.
For all $n<\omega$ and $u\in C_{[F]}$, we put $f_n(u)=\pi_n\varphi(u)$, i.e., we again consider the composition of the projection
$\pi_n$ and the homomorphism~$\varphi$.
Then $f_n$ is a homomorphism from $\mathcal{C}_{[F]}$ to $\mathcal{B}_n$.
By Lemma~\ref{l-3}, one of the following cases is possible:
\begin{enumerate}
\item[(a)]
we have $\mathcal{C}_{[G]}\leq f_n(\mathcal{C}_{[F]})\leq\mathcal{B}_n$ for some nonempty set $G\subseteq F_n\cup\{k\}$,
\item[(b)]
we have $f_n(\mathcal{C}_{[F]})\models\forall x\,g(g(x))=g(x)$.
\end{enumerate}
In case~(a), we have $G=F_n\cup\{k\}$ whence $|B_n|=2[G]+1\leqslant 2[F]+1$.
Since $\mathcal{C}_{[F]}$ is a finite algebra and $|B_n| <|B_{n+1}|$ for each $n<\omega$,
there is $s<\omega$ such that $2[F]+1<|B_n|$ for all $n\geqslant s$.
Therefore, $f_n(\mathcal{C}_{[F]})\models\forall x\,g(g(x))=g(x)$ for each $n\geqslant s$.
As $\mathcal{B}\cong{\varprojlim}\,_{j\geqslant s}\,\mathcal{B}_j$, we conclude that
$\varphi(\mathcal{C}_F)\models\forall x\,g(g(x))=g(x)$.
Therefore, the~conclusion of $\varphi^I_F$ also holds in $\mathcal{B}$ under $\gamma$
and $\mathcal{B}\models\varphi^I_F$.
\end{scproof}

\begin{claim}\label{C:cc}
There are elements $c_1,c_2\in B$ such that $\bigl(\pi_n(c_1),\pi_n(c_2)\bigr)\in\ker\beta_n{\setminus}\ker\alpha_n$
and $\pi_n(c_1),\pi_n(c_2)\in A_n$ for all $n<\omega$.
\end{claim}

\begin{scproof}
As in the proof of Claim \ref{C:4} (see the proof of Theorem \ref{T:1dg}), one can establish the existence of elements
$c_1,c_2\in B$ such that $\bigl(\pi_n(c_1),\pi_n(c_2)\bigr)\in\ker\beta_n{\setminus}\ker\alpha_n$ for all $n<\omega$.
It follows from Lemma \ref{l-3}(i) that one can choose $c_1,c_2\in B$ so that $\pi_n(c_1),\pi_n(c_2)\in A_n$ for all $n<\omega$.
\end{scproof}

\begin{claim}\label{C:4u}
The algebra $\mathcal{B}$ is not profinite with respect to $\mathbf{K}_I$.
\end{claim}

\begin{scproof}
By Lemma~\ref{l-3}, the algebra $\mathcal{B}_n$ is a subdirect product of the algebras $\mathcal{A}$ and $\mathcal{A}_n$.
As above, we denote the corresponding projections (which are onto homomorphisms) by
$\alpha_n\colon\mathcal{B}_n\to\mathcal{A}$ and $\beta_n\colon\mathcal{B}_n\to\mathcal{A}_n$.
According to Claim~\ref{C:cc},
there is a pair $\overline{c}=(c_1,c_2)\in B^2$ such that $\bigl(\pi_n(c_1),\pi_n(c_2)\bigr)\in\ker\beta_n{\setminus}\ker\alpha_n$
and $\pi_n(c_1),\pi_n(c_2)\in A_n$ for all $n<\omega$.

Suppose that $\mathcal{B}\cong{\varprojlim}\,_{t\in T}\,\mathcal{U}_t$ and $\mathcal{U}_t\in\mathbf{K}_I$ is a finite unary algebra for every $t\in T$.
Then $\mathcal{B}\leq_s\prod_{t\in T}\mathcal{U}_t$. Let $\pi^\prime_t\colon\mathcal{B}\to\mathcal{U}_t$
denote the canonical projection for every $t\in T$.
It follows from Lemma~\ref{3.2} that, for each $t\in T$, there exist $n(t)<\omega$ and a homomorphism
$\psi_t\colon\mathcal{B}_{n(t)}\to\mathcal{U}_t$ such that $\pi^\prime_t=\psi_t\pi_{n(t)}$.
Since $\pi^\prime_t$ is an onto homomorphism, we conclude that $\psi_t$ is also onto.
It follows from Lemma~\ref{l-3} that the following two cases are possible:
\begin{enumerate}
\item[(a)]
there is a nonempty set $G_t\subseteq\{k\}\cup F_{n(t)}$ such that $\mathcal{C}_{[G_t]}\leq\psi_t(\mathcal{B}_{n(t)})=\mathcal{U}_t\in\mathbf{K}_I$
and $U_t=\psi_t(B_{n(t)})=\{0\}\cup A_{[G_t]}\cup\psi_t(B_{n(t)})$, where $A_{[G_t]}=\psi_t(A_{n(t)})$;
\item[(b)]
we have $\psi_t(\mathcal{B}_{n(t)})\models\forall x\,g(g(x))=g(x)$.
\end{enumerate}
Since $\mathcal{C}_{[G_t]}\in\mathbf{K}_I$ in case (a), we conclude that $G_t\subseteq(\{k\}\cup F_{n(t)})\cap I=F_{n(t)}$ by Claim~\ref{C:1u}.
Since $\mathcal{B}\not\models\forall x\,g(g(x))=g(x)$ by Claim \ref{C:g}, the set
\[
T_0=\bigl\{t\in T\mid\mathcal{C}_{[G_t]}\leq\psi_t(\mathcal{B}_{n(t)})\ \text{for some nonempty}\ G_t\subseteq F_{n(t)}\bigr\}
\]
is coinitial in $T$. This implies that $\mathcal{B}\cong{\varprojlim}\,_{t\in T_0}\,\mathcal{U}_t$ whence $\mathcal{B}\leq_s\prod_{t\in T_0}\mathcal{U}_t$.

For each $t\in T_0$, we can find a~homomorphism $\vartheta_t\colon\mathcal{A}_{n(t)}\to\mathcal{C}_{[G_t]}\leq\mathcal{U}_t$ such that $\vartheta_t\beta_{n(t)}(a)=\psi_t(a)$
for every $a\in A_{n(t)}$.
This yields for each $i\in\{1,2\}$:
\[
\pi^\prime_t(c_i)=\psi_t\pi_{n(t)}(c_i)=\vartheta_t\beta_{n(t)}\pi_{n(t)}(c_i).
\]
Inclusion $\bigl(\pi_{n(t)}(c_1),\pi_{n(t)}(c_2)\bigr)\in\ker\beta_{n(t)}$ implies that
$(c_1,c_2)\in\ker\pi^\prime_t$ for every $t\in T_0$.
As $\mathcal{B}\leq_s\prod_{t\in T_0}\mathcal{U}_t$, we have
$c_1=c_2$ in $\mathcal{B}$, which contradicts Claim~\ref{C:cc}.
\end{scproof}

By Claims~\ref{C:2u} and~\ref{C:4u}, the quasivariety~$\mathbf{K}_I$ is not standard.

In view of Claim~\ref{C:1u}, the quasivarieties of the form~$\mathbf{K}_I$ are pairwise distinct.
It remains to refer to Proposition \ref{P:ua}.
\end{proof}

\begin{thm}\label{T:2ua}
There exist continuum many subquasivarieties of $\mathbf{V}$ that are not standard
and have an independent quasi-equational basis
$[$a directed quasi-equational basis, respectively$]$.
\end{thm}

\begin{proof}
These arguments are similar to the proofs of Theorems~\ref{T:2dg} and~\ref{T:1ua}.

We consider an infinite set $I=\{i_n\mid n<\omega\}$ and assume that its members are ordered in the natural way, i.e.,
$i_n\leqslant i_m$ if and only if $n\leqslant m$ for all $m,n<\omega$.
We put $F_{-1}=\varnothing$, $F_n=\{i_0,\ldots,i_n\}$, and $\mathcal{B}_n=\mathcal{C}_{[F_n]}$ for every $n<\omega$.

For each $m<\omega$, let $\xi_m$ denote the quasi-identity $q([F_m],[F_{m-1}])$, i.e.,
\begin{multline*}
\forall x\,\forall y_0\dots\forall y_{[F_m]-1}\,\forall z_0\dots\forall z_{[F_m]-1}\,
\bigl[f(x)=x\,\wedge\, g(x)=x\,\wedge
\\
\wedge \bigwedge_{0\leqslant i<[F_m]} f(y_i)=x\,\wedge\,\bigwedge_{0\leqslant i<[F_m]} g(y_i)=x\,
\wedge\,\bigwedge_{0\leqslant i<[F_m]} g(z_i)=y_i\,\wedge
\\
\wedge \bigwedge_{0\leqslant i<[F_m]-1} f(z_i)=y_{i+1}\,\wedge\,f(z_{[F_m]-1})=y_0\bigr]
\longrightarrow
g(z_0)=y_{[F_{m-1}]}.
\end{multline*}
We denote
\[
\Xi_I=\{\xi_m\mid m<\omega\}\quad
\text{and}\quad\mathbf{M}_I=\operatorname{Mod}\Xi_I\cap\mathbf{V}.
\]

\setcounter{claim}{0}
\begin{claim}\label{C:21u}
The set~$\Xi_I$ forms an independent quasi-equational basis of $\mathbf{M}_I$ within $\mathbf{V}$.
The quasivariety $\mathbf{M}_I$ consists of all structures $\mathcal{A}\in\mathbf{V}$ with the following property{\rm :}
\[
\text{if~} F\in\mathcal{P}_{\mathrm{fin}}(\omega) \text{~is nonempty and~} \mathcal{C}_{[F]}\in\mathbf{S}(\mathcal{A}) \text{~then~} F\nsubseteq I.
\]
\end{claim}

\begin{scproof}
It is not difficult to see that $\mathcal{B}_n$ satisfies $\xi_m$ if $m\ne n$ and violates $\xi_n$.
Hence, $\Xi_I$ is an independent set of quasi-identities.

If $\mathcal{C}_{[F]}$ is embeddable into $\mathcal{A}$ for no $F\in\mathcal{P}_{\mathrm{fin}}(\omega)$ with $\varnothing\ne F\subseteq I$
then, for every $m<\omega$, the premise of the quasi-identity $\xi_m$
holds in $\mathcal{A}$ under no interpretation $\gamma\colon\{x,y_0,\dots,y_{[F]-1},z_0,\dots,z_{[F]-1}\}\to A$ in view of Lemma~\ref{l-3}.
Therefore, $\mathcal{A}\models\xi_m$ for all $m<\omega$ and $\mathcal{A}\in\mathbf{M}_I$.
Conversely, let $\mathcal{A}\in\mathbf{M}_I$ and let $\mathcal{C}_{[F]}$ embed into $\mathcal{A}$ for some nonempty set
$F\in\mathcal{P}_{\mathrm{fin}}(\omega)$ with $F\subseteq I$. Let $i_n$ be the greatest element of $F$; then
$F\subseteq F_n$.
Therefore, the algebra $\mathcal{C}_{[F]}$ satisfies the premise of $\xi_n$
under the interpretation $\gamma$ with
\[
\gamma(x)=0,\quad\gamma(y_i)=a^{[F]}_{i(\mathrm{mod}\,[F])},\quad\gamma(z_i)=b^{[F]}_{i(\mathrm{mod}\,[F])},\quad 0\leqslant i<[F_n],
\]
but violates the conclusion of $\xi_n$ under the same interpretation $\gamma$,
see Lemma~\ref{l-3}. Hence, $\mathcal{A}\not\models\xi_n$ which contradicts our assumption $\mathcal{A}\in\mathbf{M}_I$.
\end{scproof}

As above, Claim~\ref{C:21u} also provides us with a directed quasi-equational basis of $\mathbf{M}_I$ within~$\mathbf{V}$.
Namely, for each nonempty set $F\in\mathcal{P}_{\mathrm{fin}}(\omega)$ with $F\subseteq I$, we introduce the following quasi-identity $\chi_F$:
\begin{multline*}
\forall x\,\forall y_0\dots\forall y_{[F]-1}\,\forall z_0\dots\forall z_{[F]-1}\,\forall y\,
\bigl[f(x)=x\,\wedge\, g(x)=x\,\wedge
\\
\wedge \bigwedge_{0\leqslant i<[F]} f(y_i)=x\,\wedge\,\bigwedge_{0\leqslant i<[F]} g(y_i)=x\,
\wedge\,\bigwedge_{0\leqslant i<[F]} g(z_i)=y_i\,\wedge
\\
\wedge \bigwedge_{0\leqslant i<[F]-1} f(z_i)=y_{i+1}\,\wedge\,f(z_{[F]-1})=y_0\bigr]
\longrightarrow
x=y.
\end{multline*}
We consider the set $X=\{\chi_F\mid F\in\mathcal{P}_{\mathrm{fin}}(\omega)\text{~and~} \varnothing\ne F\subseteq I\}$.

\begin{claim}\label{C:dir-u}
The set $X$ forms a directed quasi-equational basis of $\mathbf{M}_I$ within~$\mathbf{V}$.
\end{claim}

We construct an inverse spectrum $\Lambda=\langle\omega,\mathcal{B}_j,\pi_{ij}\rangle$ as follows.
According to Lemma~\ref{l-3}, for all $i,j<\omega$ with $i<j$, there is a surjective homomorphism
$\pi_{ij}\colon\mathcal{B}_j\to\mathcal{B}_i$; moreover, we may assume that $\pi_{ij}(0)=0$, $\pi_{ij}(a^{[F_j]}_k)=a^{[F_i]}_l$,
and $\pi_{ij}(b^{[F_j]}_k)=b^{[F_i]}_l$, where $k\equiv l\pmod{[F_i]}$.
According to the definition of an inverse spectrum, let $\pi_{ii}$ be the identity automorphism of~$\mathcal{B}_i$
for each $i<\omega$.

We immediately obtain the following assertion.

\begin{claim}\label{C:22u}
The triple $\Lambda=\langle\omega,\mathcal{B}_j,\pi_{ij}\rangle$ is a surjective inverse spectrum.
\end{claim}

We put $\mathcal{B}=\varprojlim\Lambda$.
Since each $\pi_{ij}$ fixes~$0$, we conclude that $B\ne\varnothing$.
Moreover, we have $\mathcal{B}\in\mathbf{S}\mathbf{P}(\mathbf{V})\subseteq\mathbf{V}$.
For every $n<\omega$, we denote by $\pi_n$ the restriction of the $n$th projection $\prod_{n<\omega}\mathcal{B}_n\to\mathcal{B}_n$ to~$\mathcal{B}$.
As $\pi_n$ is onto for every $n<\omega$, the following statement is true.

\begin{claim}\label{C:gg}
$\mathcal{B}\not\models\forall x\,g(g(x))=g(x)$.
\end{claim}

\begin{claim}\label{C:23u}
The algebra~$\mathcal{B}$ is infinite and belongs to~$\mathbf{M}_I$.
\end{claim}

\begin{scproof}
The same arguments as in the proof of Theorem \ref{T:2dg} (see Claim~\ref{C:2u} there) show that $B$ is infinite.

We prove that $\mathcal{B}\in\mathbf{M}_I$. In view of Claim \ref{C:dir-u}, it suffices to prove that $\mathcal{B}\models X$.
Assume that $F\in\mathcal{P}_{\mathrm{fin}}(\omega)$ and $\varnothing\ne F\subseteq I$.
If the premise of $\chi_F$ holds in~$\mathcal{B}$ under an interpretation $\gamma\colon\{x,y_0,\dots,y_{[F]-1},z_0,\dots,z_{[F]-1}\}\to B$
then there exists a homomorphism $\varphi\colon\mathcal{C}_{[F]}\to\mathcal{B}$ such that $\varphi(0)=0$,
$\varphi(a^{[F]-1}_k)=\gamma(y^{[F]-1}_k)$, and $\varphi(b^{[F]-1}_k)=\gamma(z^{[F]-1}_k)$ for $0\leqslant k<[F]$.

For all $n<\omega$ and $u\in C_{[F]}$, we put $f_n(u)=\pi_n\varphi(u)$.
We find that $f_n$ is a homomorphism from $\mathcal{C}_{[F]}$ to $\mathcal{B}_n$.
By Lemma~\ref{l-3}, one of the following cases occurs:
\begin{enumerate}
\item[(a)]
we have $\mathcal{C}_{[G]}\leq f_n(\mathcal{C}_{[F]})\leq\mathcal{B}_n$ for some nonempty set $G\subseteq F$,
\item[(b)]
we have $f_n(\mathcal{C}_{[F]})\models\forall x\,g(g(x))=g(x)$.
\end{enumerate}
In case (a), we have $F_n=G\subseteq F$ whence $|B_n|=2[G]+1\leqslant 2[F]+1$.
Since $\mathcal{C}_{[F]}$ is a finite algebra and $|B_n| <|B_{n+1}|$ for each $n<\omega$,
there is $s<\omega$ such that $2[F]+1<|B_n|$ for all $n\geqslant s$.
Therefore, $f_n(\mathcal{C}_{[F]})\models\forall x\,g(g(x))=g(x)$ for all $n\geqslant s$.
As $\mathcal{B}\cong{\varprojlim}\,_{j\geqslant s}\,\mathcal{B}_j$, we conclude that $\mathcal{B}\models\forall x\,g(g(x))=g(x)$, which contradicts
Claim~\ref{C:gg}.
Therefore, the~premise of $\chi_F$ holds in $\mathcal{B}$ under no interpretation $\gamma$ and $\mathcal{B}\models\chi_F$.
\end{scproof}

\begin{claim}\label{C:24u}
The algebra~$\mathcal{B}$ is not profinite with respect to~$\mathbf{M}_I$.
\end{claim}

\begin{scproof}
Consider an arbitrary homomorphism $\varphi\colon\mathcal{B}\to\mathcal{D}$, where $\mathcal{D}\in\mathbf{M}_I$ is a finite structure.
By Lemma \ref{3.2}, there is $n<\omega$ and a homomorphism $\psi\colon\mathcal{B}_n\to\mathcal{D}$ such that $\varphi=\psi\pi_n$.
By Lemma~\ref{l-3}, one of the following cases occurs:
\begin{enumerate}
\item[(a)]
we have
$\mathcal{C}_{[G]}\leq\psi(\mathcal{B}_n)\in\mathbf{M}_I$ for some nonempty set $G\subseteq F_n$;
%moreover, $\psi(B_n)=\{0\}\cup A_{[G]}\cup B^\prime$ where $f(B^\prime),g(B^\prime)\subseteq A_{[G]}$;
\item[(b)]
we have
$\psi(\mathcal{B}_n)\models\forall x\,g(g(x))=g(x)$.
\end{enumerate}
Since $\mathcal{C}_{[G]}\in\mathbf{M}_I$ in case~(a) and $\varnothing\ne G\subseteq F_n\subseteq I$,
we arrive at a contradiction with Claim~\ref{C:dir-u}. Therefore, case (a) is impossible.
Thus, we conclude that case (b) takes place and
$\varphi(\mathcal{B})=\psi(\mathcal{B}_n)\models\forall x\,g(g(x))=g(x)$.
Hence if the~unary algebra $\mathcal{B}$ were profinite with respect to $\mathbf{M}_I$, it would satisfy the identity $\forall x\,g(g(x))=g(x)$
which contradicts Claim \ref{C:gg}.
\end{scproof}

By Claims \ref{C:23u} and \ref{C:24u}, the quasivariety $\mathbf{M}_I$ is not standard.

It remains to notice that, in view of Claim~\ref{C:21u}, the quasivarieties of the form~$\mathbf{M}_I$
are pairwise distinct.
\end{proof}

For quasivarieties possessing finite B-classes, similar results on the existence of directed quasi-equational bases
(cf. Claims~\ref{C:dir-u} in the~proofs of Theorems~\ref{T:2dg} and~\ref{T:2ua}) were established in~\cite{Kr22}.

\begin{thebibliography}{88}

\bibitem{BMN}
A.~O.~Basheyeva, M.~Mustafa,  A.~M.~Nurakunov,
\emph{Properties not retained by pointed enrichments of finite lattices},
Algebra Universalis \textbf{81}~(2020), Art.~56. 

\noindent
[https://doi.org/10.1007/s00012-020-00692-4].

\bibitem{BNSZ}
A.~Basheyeva, A.~M.~Nurakunov, M.~V.~Schwidefsky, A.~Zamojska-Dzienio,
{\it Lattices of sub\-class\-es.}~III,
Siberian Electron. Math. Reports \textbf{14} (2017), 252--263

\noindent
[http://semr.math.nsc.ru/v14/p252-263.pdf].

\bibitem{CD}
D.~M. Clark, B.~A. Davey,
{\it Natural Dualities for the Working Algebraist},
Cambridge University Press, Cambridge, 1998.

\bibitem{CDF}
D.~M.~Clark, B.~A.~Davey, R.~S.~Freese, M.~Jackson,
\emph{Standard topological algebras: syntactic and principal congruences and profiniteness},
Algebra Universalis \textbf{52} (2005), 343--376

\noindent
[https://doi.org/10.1007/s00012-004-1917-6].

\bibitem{CDH}
D.~M.~Clark, B.~A.~Davey, M.~Haviar, J.~G.~Pitkethly, M.~R.~Talukder,
\emph{Standard topological quasi-varieties},
Houston J.~Math. \textbf{29} (2003), 859--887.

%\noindent
%[].

\bibitem{DCJ}
D.~M.~Clark, B.~A. Davey, M.~G. Jackson, J.~G. Pitkethly,
{\it The axiomatizability of topo\-log\-i\-cal prevarieties},
Adv. Math. \textbf{218}~(2008), 1604--1653

\noindent
[https://doi.org/10.1016/j.aim.2008.03.020].

\bibitem{DJMM}
B.~A.~Davey, M.~Jackson, M.~Maroti, R.~N.~McKenzie,
\emph{Principal and syntactic congruences in congruence-distributive and congruence-permutable varieties},
J.~Austral. Math. Soc. \textbf{85} (2008), 59--74

\noindent
[https://doi.org/10.1017/S144678870800061X].

\bibitem{G}
V.~A. Gorbunov,
{\it Algebraic Theory of Quasivarieties},
Plenum, New York, 1998.

\bibitem{GoKr01}
V.~A. Gorbunov, A.~V. Kravchenko,
{\it Universal Horn classes and colour-families of graphs},
Algebra Universalis \textbf{46} (2001), 43--67

\noindent
[https://link.springer.com/article/10.1007/PL00022468].

\bibitem{J}
P.~T. Johnstone,
{\it Stone Spaces},
Cambridge University Press, Cambridge, 1982.

\bibitem{Kr02}
A.~V.~Kravchenko,
{\it Complexity of quasivariety lattices for varieties of unary algebras},
Siberian Adv.\ Math. \textbf{12} (2002), 63--76

\noindent
[https://doi.org/10.3103/S105513440903002X].

\bibitem{Kr08}
A.~V. Kravchenko,
{\it On the lattices of quasivarieties of differential groupoids},
Comment. Math. Univ. Carolin. \textbf{49} (2008), 11--17

\noindent
[https://cmuc.karlin.mff.cuni.cz/pdf/cmuc0801/krav.pdf].

\bibitem{Kr16}
A.~V.~Kravchenko,
{\it Complexity of quasivariety lattices for varieties of unary algebras.}~II,
Siberian Electron. Math. Reports \textbf{13} (2016), 388--394~(in Russian)

\noindent
[https://doi.org/10.17377/semi.2016.13.034].

\bibitem{Kr22}
A.~V. Kravchenko,
{\it On directed and finitely partitionable bases for quasi-identities},
Siberian Electron. Math. Reports \textbf{19} (2022), submitted.

%\noindent
%[https://].

\bibitem{DG2}
A.~V. Kravchenko, M.~V. Schwidefsky,
{\it On the complexity of the lattices of subvarieties and congruences.} II.
{\it Differential groupoids and unary algebras},
Siberian Electron. Math. Reports \textbf{17} (2020), 753--768

\noindent
[http://semr.math.nsc.ru/v17/p753-768.pdf].

\bibitem{DG}
A.~V. Kravchenko, A.~M. Nurakunov, M.~V. Schwidefsky,
{\it Quasi-equational bases of dif\-fer\-en\-tial group\-o\-ids and unary algebras},
Siberian Electron. Math. Reports \textbf{14} (2017), 1330--1337

\noindent
[http://semr.math.nsc.ru/v14/p1330-1337.pdf].

\bibitem{KrNuSchw18}
A.~V. Kravchenko, A.~M. Nurakunov, M.~V. Schwidefsky,
{\it Structure of quasivariety lattices.} I. {\it  Independent
axiomatizability},
Algebra and Logic \textbf{57} (2018), 445--462

\noindent
[https://link.springer.com/article/10.1007/s10469-019-09516-4].

\bibitem{KrNuSchw19}
A.~V. Kravchenko, A.~M. Nurakunov, M.~V. Schwidefsky,
{\it Structure of quasivariety lattices.} II. {\it Undecidable
problems},
Algebra and Logic \textbf{58} (2019), 123--136

\noindent
[https://link.springer.com/article/10.1007/s10469-019-09531-5].

\bibitem{KrNuSchw20}
A.~V. Kravchenko, A.~M. Nurakunov, M.~V. Schwidefsky,
{\it Structure of quasivariety lattices.} III. {\it Finitely
partitionable bases},
Algebra and Logic \textbf{59} (2020), 222--229

\noindent
[https://link.springer.com/article/10.1007/s10469-020-09594-9].

\bibitem{KrNuSchw21}
A.~V. Kravchenko, A.~M. Nurakunov, M.~V. Schwidefsky,
{\it Structure of quasivariety lattices.} IV. {\it Nonstandard quasivarieties},
Siberian Math.~J. \textbf{62} (2021), 850--858

\noindent
[https://link.springer.com/article/10.1134/S0037446621050074].

\bibitem{NS}
A.~M.~Nurakunov, M.~M.~Stronkowski,
\emph{Profiniteness in finitely generated varieties is undecidable},
J.~Symbolic Logic \textbf{83} (2018), 1566--1578

\noindent
[https://doi.org/10.1017/jsl.2017.89].

\bibitem{Za}
L.~Ribes, P.~Zalesskii,
{\it Profinite Groups},
Springer Verlag, Berlin, 2000.

\bibitem{RoRo87}
A.~Romanowska, B.~Roszkowska,
{\it On some groupoid modes},
Demonstr. Math. \textbf{20} (1987), 277--290

\noindent
[https://www.degruyter.com/document/doi/10.1515/dema-1987-1-224/pdf].

\bibitem{RoSm02}
A.~B.~Romanowska, J.~D.~H.~Smith,
{\it Modes},
World Scientific, Singapore, 2002.

\bibitem{Sch15}
M.~V.~Schwidefsky,
{\it Complexity of quasivariety lattices},
Algebra and Logic \textbf{54} (2015), 245--257

\noindent
[https://doi.org/10.1007/s10469-015-9344-7].

\bibitem{SZ2}
M.~V.~Schwidefsky, A.~Zamojska-Dzienio,
{\it Lattices of subclasses.} II,
Internat. J.~Algebra Comput. \textbf{24} (2014), 1099--1126.

\noindent
[https://doi.org/10.1142/S0218196714500489].

\bibitem{T}
B. Trotta,
\emph{Residual properties of reflexive anti-symmetric digraphs},
Houston J.~Math. \textbf{37} (2011), 27--46

\noindent
[https://doi.org/10.1007/s00012-010-0097-9].

\end{thebibliography}

\end{document}
