\documentclass[10pt,draft]{amsart}
\usepackage[cp1251]{inputenc}
\usepackage[english,russian]{babel}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsfonts}
\usepackage{graphicx}

\def\udcs{512.57} %Here the author places classificators of the paper according to Russian classification system
\def\mscs{08C15} %Here the author places  classificators according to the AMS classification list.
\setcounter{page}{144}

\newtheorem{theorem}{Theorem}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}

\theoremstyle{definition}
\newtheorem{definition}{Definition}

\def\logo{{\bf\huge S\raisebox{0.2ex}{\hspace{0.55ex}\raisebox{0.05ex}e\hspace{-1.65ex}$\bigcirc$}MR}}

\def\semrtop
     {
  \vbox{
     \noindent\logo
     \hspace{80mm}\raisebox{1ex}{ISSN 1813-3304 }

     \vspace{5mm}

     \begin{center}
     {\huge СИБИРСКИЕ \ ЭЛЕКТРОННЫЕ} \\[2mm]  %Headings in Russian
     {\huge МАТЕМАТИЧЕСКИЕ ИЗВЕСТИЯ} \\[2mm]
     {\large Siberian Electronic Mathematical Reports} \\[1mm]
     {\LARGE\tt{http://semr.math.nsc.ru}}\\[0.5mm]
%     {\small Институт математики им. С.Л.Соболева СО РАН}\\[-1mm]
%     {\small Sobolev Institute of Mathematics SB RAS}
     \end{center}
     \vspace{-3mm}
     \noindent
     \begin{tabular}{c}
     \hphantom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa} \\
     \hline\hline
     \end{tabular}

     \vspace{1mm}
     {\flushleft\it Том 16, стр. 144--144 (2019) \hspace{65mm}{\rm\small УДК \udcs}} %Volume, pages in Russian,
                                                                                    %filled by Editorial board
     \newline
         {\rm\small DOI~10.33048/semi.2019.16.xxx}\hphantom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa}{\rm\small MSC\ \ \mscs }
  }%\vbox
}

\begin{document}
\renewcommand{\refname}{References}
\renewcommand{\proofname}{Proof.}

\thispagestyle{empty}

\title[Bases of quasi-identities]{On directed and finitely partitionable bases for quasi-identities}
\author{{A. V. Kravchenko}}%
\address{Aleksandr Vladimirovich Kravchenko
\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}%

\thanks{\sc Kravchenko, A.~V.,
On directed and finitely partitionable bases for quasi-identities}
\thanks{\copyright \ 2022 Kravchenko, A.~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 May 3, 2022, published ???.}%


\semrtop \vspace{1cm}
\maketitle {\small
\begin{quote}
\noindent{\sc Abstract. } We prove that, under certain conditions on a~quasivariety, there exists continuum many subquasivarieties of this quasivariety with both finitely partitionable (independent) and directed bases for quasi-identities. We also notice that such a~situation is impossible for bases for anti-identities.\medskip

\noindent{\bf Keywords:} quasivariety, basis for quasi-identities.
 \end{quote}
}


\section{Introduction}

In~\cite{KrNuSchw18}, the notion is introduced of a B-class.
Existence of such a class with respect to a~quasivariety~$\mathbf{K}$ witnesses a very complicated structure of~$\mathbf{K}$. A~series of examples is listed in~\cite{KrNuSchw19}. They include quasivarieties of well-known structures with no independent basis for their quasi-identities. As is shown in~\cite{KrNuSchw20}, the~intersection of a~family of such subquasivarieties admits a~finitely partitionable (independent) basis for its quasi-identities. 

In~\cite{GoKr01}, the~notion is introduced of a~directed basis
for sentences of a~class of structures; moreover, conditions are
indicated that are equivalent to existence of such a~basis for
anti-identities. The~notions of directed and independent bases are quite opposite; namely, a~basis is both directed and independent if and only if it is a~singleton. 
However, existence of a~B-class with respect to a~quasivariety~$\mathbf{K}$ implies existence of continuum many subquasivarieties of~$\mathbf{K}$ admitting a~finitely partitionable basis and (another) directed basis for its quasi-identities. We also notice that such a~situation is impossible for anti-identities.  

\section{Preliminaries}

Throughout the~article, we consider structures and classes of structures of a~fixed finite similarity type~$\sigma$.

Recall that a~{\it quasi-identity\/} is a~universal Horn sentence of the~form
$$
\forall\overline{x}\,
\bigg(\big(\varphi_1(\overline{x})\&\dots\&\varphi_n(\overline{x})\big)
\to\varphi_0(\overline{x})\bigg)
$$
and an~{\it anti-identity\/} is a~(negative) universal Horn sentence of the~form
$$
\forall\overline{x}\,
\big(\neg\varphi_1(\overline{x})\vee\dots\vee\neg\varphi_n(\overline{x})\big),
$$ 
where each $\varphi_i$ is an~atomic formula with free variables among the~entries of~$\overline{x}$.
A~{\it quasivariety\/} ({\it antvariety\/}) is a~class of structures defined by a~set of quasi-identities (anti-identities).

Let $\mathbf{K}\subseteq\mathbf{M}$, where $\mathbf{K}$ and $\mathbf{M}$ are universal Horn classes. 

\begin{definition}\label{b-ind}
A~set $\Phi$ of quasi-identities (anti-identities) is a~{\it finitely paritionable basis\/} for quasi-identities (anti-identities) of $\mathbf{K}$ in $\mathbf{M}$ if there is a~countable set~$J$ and a~partition 
$\Phi=\bigcup_{n\in J}\Phi_n$ such that $\Phi_n$ is finite for every $n\in J$, we have $\mathbf{K}=\mathbf{M}\cap\operatorname{Mod}(\Phi)$, and the~condition
$\mathbf{K}\ne\mathbf{M}\cap\operatorname{Mod}\bigl(\Phi{\setminus}\Phi_n\bigr)$ holds for every $n\in J$.
\end{definition}

Notice that no block $\Phi_n$ can be empty.
If $\Phi$ is a~finitely partitionable basis with $|\Phi_n|=1$ for every $n\in J$ then $\Phi$ is said to be {\it independent}.

Notice that our definition slightly differs from the generally accepted one (see, for example,~\cite{Ma67}). 
Namely, infinite bases are usually considered and 
the~set of natural numbers is taken for~$J$. Consideration of both infinite and finite bases allows us to establish a~more 
con\-ve\-nient connection between properties of a~basis (see Proposition~\ref{d-i}).

The~following notion was introduced in~\cite{GoKr01}.

\begin{definition}\label{b-dir}
A set $\Phi=\{\varphi_i: i\in I\}$
of quasi-identities (anti-identities) is {\it directed in\/}
$\mathbf{M}$ if, for all $i,j\in I$, there exists $k\in I$
such that $\varphi_i$ and $\varphi_j$ are consequences of $\varphi_k$
in the class $\mathbf{M}$. If $\Phi$ is a directed in $\mathbf{M}$ set of quasi-identities (anti-identities) and $\mathbf{K}=\mathbf{M}\cap\operatorname{Mod}(\Phi)$ then we say that $\Phi$ is a~{\it directed basis\/} for quasi-identities (anti-identities) of $\mathbf{K}$ in~$\mathbf{M}$.
\end{definition}

The~following fact is immediate from Definitions~\ref{b-ind} and~\ref{b-dir}.

\begin{proposition}\label{d-i}
If a~basis~$\Phi$  is simultaneously directed and finitely partitionable then it is finite. 
In particular, if~$\Phi$ is simultaneously directed and independent then it is a~singleton.
\end{proposition}

\begin{proof}
Consider two representations of~$\Phi$. 
Since it is finitely partitionable, we have $\Phi=\bigcup_{n\in J}\Phi_n$, where each $\Phi_n$ is finite. 
Since it is directed, for every $n\in J$, there exists $i_n\in I$ such that every sentence in $\Phi_n$ 
is a~consequence of~$\varphi_{i_n}$ in~$\mathbf{M}$. 
If $\varphi_{i_n}\notin\Phi_n$ then 
$\mathbf{M}\cap\operatorname{Mod}(\Phi)=\mathbf{M}\cap\operatorname{Mod}\bigl(\Phi{\setminus}\Phi_n\bigr)$, 
which is a~contradiction.
If $|J|>1$ then we consider $n,m\in J$ with $n\ne m$ and find $k\in J$ 
such that~$\varphi_{i_n}$ and~$\varphi_{i_m}$ are consequences of $\varphi_k$ in~$\mathbf{M}$. 
Since $\varphi_k\notin\Phi_n\cap\Phi_m=\varnothing$, we again arrive at a~contradiction.
We conclude that $J=1$, i.e., $\Phi$ is a~finite set (which is a~singleton if $\Phi$ is an~independent basis). 
\end{proof}

Kartashov~\cite{Ka80} found a~quasivariety of monounary algebras that admits an~infinite directed basis and another infinite independent basis for its quasi-identities. We show that, under certain conditions, a~quasivariety possesses $2^\omega$ subquasivarieties with a~similar property; however, such a~situation is impossible for anti-identities.

The~main tool is the~notion of a~B-class introduced in~\cite{KrNuSchw18}.
Let $\mathbf{M}$ be a quasivariety.
We denote by $\mathcal{P}_{\mathrm{fin}}(\omega)$ the set of finite subsets of the~set~$\omega$ of natural numbers.
We denote by $\mathbf{Q}(\mathbf{K})$ the~least quasivariety extending a~class~$\mathbf{K}$ and by $\mathbf{H}(\mathbf{K})$ the~class of homomorphic images of structures in~$\mathbf{K}$.
A class $\mathbf{A}=\{\mathcal{A}_F: F\in\mathcal{P}_{\mathrm{fin}}(\omega)\}\subseteq\mathbf{M}$ is called a B-{\it class with respect to\/} $\mathbf{M}$ if $\mathbf{A}$ satisfies the following conditions:
\begin{enumerate}
\item[$(\mathrm{B}_0)$]
for every nonempty $F\in\mathcal{P}_{\mathrm{fin}}(\omega)$, the structure $\mathcal{A}_F$ is finitely presented in $\mathbf{M}$;
the~structure $\mathcal{A}_\varnothing$ is trivial;
\item[$(\mathrm{B}_1)$]
if $F=G\cup H$ in $\mathcal{P}_{\mathrm{fin}}(\omega)$
then $\mathcal{A}_F\in\mathbf{Q}(\mathcal{A}_G,\mathcal{A}_H)$;
\item[$(\mathrm{B}_2)$]
if $F,G\in\mathcal{P}_{\mathrm{fin}}(\omega)$, $F\ne\varnothing$, and $\mathcal{A}_F\in\mathbf{Q}(\mathcal{A}_G)$
then $F=G$;
\item[$(\mathrm{B}_3)$]
if $F\in\mathcal{P}_{\mathrm{fin}}(\omega)$, $i\in\omega$, and
$f$ is a~homomorphism from $\mathcal{A}_F$ to~$\mathcal{A}_{\{i\}}$ 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}

As is shown in~\cite{KrNuSchw18,KrNuSchw19,KrNuSchw20,KrNuSchw21}, existence of a~B-class with respect to a~quasivariety~$\mathbf{M}$ witnesses complexity of~$\mathbf{M}$ from many points of view. Namely, such a~quasivariety is $\mathcal{Q}$-universal, there exists a~subclass $\mathbf{K}\subseteq\mathbf{M}$ such that the~set of (isomorphism types) of finite sublattices of the lattice $\mathrm{L_q}(\mathbf{K})$ of its relative subquasivarieties is undecidable, and there exist $2^\omega$ elements of $\mathrm{L_q}(\mathbf{M})$ with no covers, $2^\omega$ subquasiva\-ri\-eties of~$\mathbf{M}$ with no finitely partitionable bases for their quasi-identities, and $2^\omega$ nonstandard subquasivarieties of~$\mathbf{M}$; moreover, a~series of natural algorithmic problems is undecidable for $2^\omega$ subquasivarieties of~$\mathbf{M}$. In the~present article, we indicate one more point of view from which a~quasivariety with a~B-class is complicated.

\section{Bases of quasi-identities}

By~\cite[Theorem~8.1]{KrNuSchw20}, if there exists a~B-class with respect to a~quasivariety $\mathbf{M}$ then there exist
$2^\omega$ subquasivarieties of~$\mathbf{M}$ without finitely partitionable bases for their quasi-identities such that the~intersection admits a~finitely partitionable basis for its quasi-identities. Moreover, in~\cite[Theorem~8.2]{KrNuSchw20},
conditions are found for such a~basis to be independent. 

The~proofs of those assertions are ``relatively constructive,'' i.e., they provide us with a~description of the~intersection  of subquasivarieties in terms of the~B-class. 

We denote by $\mathcal{P}_\mathrm{inf}$ the~set of infinite subsets~$I\subseteq\omega$ such that the~complement $\omega\setminus I$ is infinite too. For every~$I\in\mathcal{P}_\mathrm{inf}$, we consider the~class~$\mathbf{K}_I$ of all structures $\mathcal{A}\in\mathbf{M}$ with the following property: The~structure $\mathcal{A}_F$ is embeddable into~$\mathcal{A}$ if and only if $F\subseteq I$.
We put 
$\mathbf{K}=\bigcap_{I\in\mathcal{P}_\mathrm{inf}}\mathbf{K}_I$.
The~following assertion is~immediate from~\cite[Claim~8.3]{KrNuSchw20}.

\begin{proposition}\label{forbid}
The~class~$\mathbf{K}$ is a~quasivariety and consists of all structures $\mathcal{A}\in\mathbf{M}$
with the following property{\rm :}
If $\mathcal{A}_F$ is embeddable into~$\mathcal{A}$ then $F=\varnothing$.
\end{proposition}

By~\cite[Claim~8.5]{KrNuSchw20}, there exists a~finitely partitionable basis for quasi-identities of~$\mathbf{K}$ in~$\mathbf{M}$. We find another basis for its quasi-identities.

\begin{proposition}
There exists a~directed basis for quasi-identities of~$\mathbf{K}$
in~$\mathbf{M}$.
\end{proposition}

\begin{proof}
Consider a nonempty set $F\in\mathcal{P}_{\mathrm{fin}}(\omega)$. According to condition $(\mathrm{B}_0)$, there are a finite set $X_F$ of variables and a finite set $\Delta_F$ of atomic formulas with free variables belonging to $X_F$ such that
$\mathcal{A}_F\cong\mathcal{F}_{\mathbf{M}}(X_F,\Delta_F)$,
i.e., the~structure $\mathcal{A}_F$ is finitely presented in~$\mathbf{M}$ by the~generators~$X_F$ and relations~$\Delta_F$.
Let $\gamma_F\colon X_F\to A_F$ be the corresponding interpretation of the variables from $X_F$.
Consider an arbitrary set $G\subseteq F$.
By~\cite[Lemma 2.2(ii)]{KrNuSchw18}, there is a homomorphism from $\mathcal{A}_F$ onto $\mathcal{A}_G$.
Then there is a finite set $\Delta^F_G$ of atomic formulas
such that $\mathcal{A}_G\cong\mathcal{F}_{\mathbf{M}}(X_F,\Delta^F_G)$ with an~interpretation $\gamma^F_G$ of the variables
from $X_F$ in $A_G$ and $\Delta^F_G\models_{\mathbf{M}}\Delta_F$
(the~reader is referred to~\cite[Sec.~1]{KrNuSchw18} for more detail).

Let $F\in\mathcal{P}_\mathrm{fin}(\omega)$. We consider the~following sentence $\psi_F$ which is equivalent to a~finite set of quasi-identities:
$$
\forall\overline{x} \left(\& \Delta_F(\overline{x})\to\&\Delta_\varnothing^F(\overline{x})\right),
$$
where the~tuple~$\overline{x}$ corresponds to the~set~$X_F$ of generators. Let $\Psi_\mathbf{K}=\{\psi_F: F\in\mathcal{P}_\mathrm{fin}(\omega)\}$.

We show that $\Psi_\mathbf{K}$ is a~directed set. For $F,G\in\Psi_\mathbf{K}$, we put $H=F\cup G$. Let $\mathcal{A}\in\mathbf{M}$ and let $\mathcal{A}$ satisfy the~sentence~$\psi_H$.
Assume that $\mathcal{A}$ satisfies the~premise $\&\Delta_F$ with an interpretation~$\gamma$ of the~variables from~$X_F$ in~$A$. Then there exists a~homomorphism~$f$ from $\mathcal{A}_F$ to~$\mathcal{A}$. We represent $\mathcal{A}_F\cong\mathcal{F}_\mathbf{M}(X_H,\Delta_F^H)$ with an interpretation $\gamma_F^H$ of variables from $X_H$ in $A_F$. Then $\mathcal{A}$ satisfies the~premise $\&\Delta_F^H$ with the~interpretation~$f\gamma_F^H$. Since $\Delta_F^H\models_\mathbf{M}\Delta_H$, we conclude that $\mathcal{A}$ satisfies the~premise $\&\Delta_H$ with the~same interpretation. Since $\mathcal{A}\models\psi_H$, the~structure~$\mathcal{A}$ satisfies the~conclusion of $\psi_H$ with the~same interpretation, i.e., the~homomorphic image of~$\mathcal{A}_H$ under the~composition of homomorphisms is a~trivial structure. This means that $f(\mathcal{A}_F)$ is a~trivial structure too, i.e., the~conclusion of $\psi_F$ holds in $\mathcal{A}$ with the~interpretation~$\gamma$. 

It remains to show that $\Psi_\mathbf{K}$ is a~basis for quasi-identities of $\mathbf{K}$ in~$\mathbf{M}$. Let $\mathcal{A}\in\mathbf{K}$ and let $\mathcal{A}$ satisfy the~premise of $\psi_F$ with an~interpretation $\gamma$ of the~variables from $X_F$ in~$A$. Then there exists a~homomorphism $f$ from $\mathcal{A}_F$ to $\mathcal{A}$. The~homomorphic image $f(\mathcal{A}_F)$ belongs to $\mathbf{H}(\mathcal{A}_F)$ and is a~substructure of~$\mathcal{A}$. We have $\mathcal{A}\in\mathbf{K}\subseteq\mathbf{M}$; hence, $f(\mathcal{A}_F)\in\mathbf{M}$. By~($\mathrm{B}_4$), we find that $f(\mathcal{A}_F)\cong\mathcal{A}_G$ for some $G\in\mathcal{P}_\mathrm{fin}(\omega)$. 
By Proposition~\ref{forbid}, we have $G=\varnothing$. Therefore, the~structure $f(\mathcal{A}_F)$ is trivial and, consequently, satisfies the~conclusion of~$\psi_F$ with the~same interpretation~$\gamma$. Conversely, if $\mathcal{A}\in\mathbf{M}$ and $\mathcal{A}\models\psi_F$ for every $F\in\mathcal{P}_\mathrm{fin}(\omega)$ then no nontrivial structure of the~form $\mathcal{A}_F$ admits a homomorphism to $\mathcal{A}$.
We conclude that $\mathcal{A}_F$ is embeddable into $\mathcal{A}$ if and only if $F=\varnothing$. It remains to apply Proposition~\ref{forbid}.
\end{proof}

We summarise the~above results as follows.

\begin{proposition}\label{theo}
If there exists a~$\mathrm{B}$-class with respect to a~quasivariety~$\mathbf{M}$ then there exist distinct bases $\Phi_\mathbf{K}$ and~$\Psi_\mathbf{K}$ for quasi-identities of a~suitable subquasivariety $\mathbf{K}\subseteq\mathbf{M}$ such that $\Phi_\mathbf{K}$ is finitely partitionable and $\Psi_\mathbf{K}$ is directed.
\end{proposition}

By~\cite[Claim~8.1]{KrNuSchw20}, existence of a~B-class with respect to~$\mathbf{M}$ 
implies existence of $2^\omega$ such classes; moreover, the~corresponding quasivarieties for distinct B-classes are distinct too. 
Hence, we take into account Proposition~\ref{theo} and obtain the~following assertion.

\begin{theorem}\label{continuum}
If there exists a~$\mathrm{B}$-class with respect to a~quasivariety~$\mathbf{M}$ then there exist $2^\omega$ subquasivarieties~$\mathbf{K}$ of~$\mathbf{M}$ such that $\mathbf{K}$ admits a~finitely partitionable basis $\Phi_\mathbf{K}$ and a~directed basis~$\Psi_\mathbf{K}$ in~$\mathbf{M}$.
\end{theorem}

An~additional condition on a~B-class guarantees that the~finitely partitionable basis for quasi-identities becomes independent, see~\cite[Theorem~8.2]{KrNuSchw20}. Thus, there exist examples of quasivarieties with (distinct) independent and directed infinite bases for their quasi-identities. The~proof of the~following assertion is immediate from~\cite[Corollary~9.1]{KrNuSchw20} and Theorem~\ref{continuum}.
 
\begin{corollary}
If~$\mathbf{X}$ is one of the~classes below then the~conclusion of Theorem~\emph{\ref{continuum}} holds for a~suitable subquasivariety $\mathbf{M}\subseteq\mathbf{X}${\rm :}
\begin{itemize}
\item
the variety $\mathbf{U}$ of monounary algebras without $1$-cycles,
\item
the variety $\mathbf{R}$ of all commutative rings with unit,
\item
the variety $\mathbf{C}_{mn}$ of all Cantor algebras, where $0<m<n<\omega$,
\item
the quasivariety $\mathbf{G}$ of all directed loopless graphs,
\item
the quasivariety $\mathbf{M}_{0,1}$ of all modular $(0,1)$-lattices,
\item
every variety of $(0,1)$-lattices that contains a finite non-distributive simple $(0,1)$-lat\-tice,
\item
every quasivariety of undirected antireflexive graphs that contains a non-bipartite graph.
\end{itemize}
In the~first four cases, the~bases $\Phi_\mathbf{K}$ are independent.
\end{corollary}

\section{Bases of anti-identities}

We prove that an~analogue of Theorem~\ref{continuum} is no longer  valid for anti-identities.
 
By~\cite[Theorem~4.8]{GoKr01}, if $\mathbf{M}$ is a~proper universal Horn class (i.e., no structure in~$\mathbf{M}$ possesses 
a~trivial substructure) and $\mathbf{K}$ is an~$\mathbf{M}$-antivariety then the following conditions are equivalent:
\begin{itemize}
\item[{\rm (1)}]
there exists a~directed basis for anti-identities of 
$\mathbf{K}$ in~$\mathbf{M}$;
\item[{\rm (2)}]
the~element $\mathbf{K}$ is meet irreducible in the~lattice
of $\mathbf{M}$-antivarieties;
\item[{\rm (3)}]
we have 
$\mathcal{A}\times\mathcal{B}\in\mathbf{M}\setminus \mathbf{K}$ for all 
$\mathcal{A},\mathcal{B}\in\mathbf{M}\setminus\mathbf{K}$.
\end{itemize}

We show that each of these equivalent conditions 
is equivalent to the~following condition:
\begin{itemize}
\item[{\rm (4)}]
each basis for anti-identities of $\mathbf{K}$ in $\mathbf{M}$
is directed.
\end{itemize}

Indeed, it is clear that~(4) yields~(1); it suffices to prove that,
say,~(3) implies~(4). Essentially, we pass to the~set of (isomorphism types) of finitely presented structures in $\mathbf{M}\setminus\mathbf{K}$ partially ordered by homomorphisms and take into account the~fact that each subset generating a~directed set is directed.

\begin{theorem}
If an~antivariety possesses a~directed basis for its anti-identities then each basis for its anti-identities is directed.
\end{theorem}

\begin{proof}
Let $\mathbf{K}$ be defined in $\mathbf{M}$ by a basis
$\Sigma=\{\varphi_i: i\in I\}$ for its anti-identities.
For every anti-identity
$$
\varphi\leftrightharpoons
\forall\overline{x}\,
\big(\neg\psi_1(\overline{x})\vee\dots\vee\neg\psi_n(\overline{x})\big),
$$
let $\mathcal{A}(\varphi)$ denote the structure defined in $\mathbf{M}$
by the~generators $\overline{x}$ and the~relations $\psi_1,\dots,\psi_n$. It is easy to see
that a~structure $\mathcal{B}\in\mathbf{M}$ satisfies an~anti-identity $\varphi$ if and only if $\mathcal{A}(\varphi)$ admits no homomorphism
to~$\mathcal{B}$.

In particular, for every $i\in I$, the anti-identity $\varphi_i$ is false
in $\mathcal{A}(\varphi_i)$, i.e., for all $i,j,\in I$, we have
$\mathcal{A}(\varphi_i),\mathcal{A}(\varphi_j)\notin\mathbf{K}$.
%; moreover, the structures $\mathcal{A}(\varphi_i)$ and $\mathcal{A}(\varphi_j)$ are finitely presented. 
By~(3), we obtain $\mathcal{A}(\varphi_i)\times\mathcal{A}(\varphi_j)\notin\mathbf{K}$.
Hence, there exists $k\in I$ such that
$\mathcal{A}(\varphi_i)\times\mathcal{A}(\varphi_j)$
does not satisfy $\varphi_k$. This means that $\mathcal{A}(\varphi_k)$
admits a homomorphism to $\mathcal{A}(\varphi_i)\times\mathcal{A}(\varphi_j)$.
Hence, $\mathcal{A}(\varphi_k)$ admits a homomorphism to both~$\mathcal{A}(\varphi_i)$ and~$\mathcal{A}(\varphi_j)$.

Let $\mathcal{B}\in\mathbf{M}$ be an arbitrary structure obeying~$\varphi_k$.
Then $\mathcal{A}(\varphi_k)$ admits no homomorphism to $\mathcal{B}$.
Hence, none of $\mathcal{A}(\varphi_i)$, $\mathcal{A}(\varphi_j)$ admits a homomorphism to $\mathcal{B}$. This means that $\mathcal{B}$
satisfies both $\varphi_i$ and $\varphi_j$. Since $\mathcal{B}\in\mathbf{M}$
is arbitrary, we conclude that $\varphi_i$ and $\varphi_j$ are consequences of
$\varphi_k$ in~$\mathbf{M}$.
\end{proof}

\begin{thebibliography}{1}


\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{Ka80}
V.~K. Kartashov,
{\it Quasivarieties of unars},
Math. Notes \textbf{27} (1980), 5--12

\noindent
[https://link.springer.com/article/10.1007/BF01149807].

\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{Ma67}
A.~I. Mal'tsev,
{\it Universally axiomatizable subclasses of locally finite classes of models},
Siberian Math. J. \textbf{8} (1967), 764--770

\noindent
[https://link.springer.com/content/pdf/10.1007/BF01040652.pdf].

\end{thebibliography}

\end{document}