In this paper the authors investigate category equivalences of particular varieties (i.e., equational classes) of finitary algebras. Since such equivalences preserve some important properties (e.g., Mal’tsev-type conditions), it is of some interest to find algebraic conditions for a variety to be equivalent (as a category) to some given variety \({\mathcal K}\). If \({\mathcal K}\) is the variety of all Boolean algebras or the variety of all distributive lattices, such conditions are well-known. In both cases \({\mathcal K}\) is generated by a two-element algebra with a so-called majority term function, i.e., a ternary term function satisfying the identities \(d(x,x,y)=d(x,y,x)=d(y,x,x)=x\). In an earlier paper [Beitr. Algebra Geom. 21, 35-56 (1986; Zbl 0619.08001)] the second author handled the case where \({\mathcal K}\) is any category generated by a two-element algebra with a majority term function.
Carrying on these investigations the authors deal with varieties generated by a two-element algebra with a near-unanimity term function, i.e., an \((n+1)\)-ary term function \(u\) on \(A\) \((n\geq 3)\) satisfying the identities \(u(x,\dots,x,y,x,\dots,x)=x\) for any position of variable \(y\).
The main results of the paper under review are contained in two theorems (the “structure theorem” and the “existence theorem”). The first one (Theorem 4.7) yields necessary conditions for arbitrary varieties to be equivalent to a variety \(V(\underline{2_ N})\) generated by a two- element algebra \(\underline{2_ N}\) with a near-unanimity term function. The second theorem (Theorem 6.3) says that such varieties equivalent to \(V(\underline{2_ N})\) actually exist.
There are a few misprints which, however, are not serious at all, maybe with one exception. In the formulation of the structure theorem (Theorem 4.7) there is a part missing, making the formulation of the theorem incomprehensible. Though it is not too hard for the reader to find out what is missing, we will give here the correct version of the theorem. We need some more definitions. For any non-empty finite set \(A\), we denote the set of all \(n\)-ary operations on \(A\) \((n\geq 1)\) by \(O_ A^{(n)}\), and stipulate \(O_ A:=\bigcup^{\infty}_{n=1}O_ A^{(n)}\). If \(\rho\subseteq A^ h\) is an \(h\)-ary relation on \(A\), \(h\geq 1\), \(\text{Pol} \rho\) is the set of all operations of \(O_ A\) preserving \(\rho\). Let \(\underline A=(A;F)\) be a finitary algebra with universe \(A\). Relation \(\rho\subseteq A^ h\) is said to be a compatible relation of \(\underline A\), if all operations of algebra \(\underline A\) preserve \(\rho\). The clone of all term functions of \(\underline A\) is denoted by \(T(\underline A)\). Let \(R\) be a set \(\{\rho_ 1,\dots,\rho_ n\}\) of relations on \(A\). We say that \(\underline A\) is generated by \(R\), if \(T(\underline A)=\bigcap^ n_{k=1}\text{Pol} \rho_ k\). We then write \(\underline{A_ R}\) rather than \(\underline A\). In case \(R\) consists of just one relation \(\rho\) we simply write \(\underline{A_{\rho}}\). The complete formulation of the structure theorem is as follows (the square brackets “[” resp. “]” indicate the part missing in the reviewed article):
Theorem 4.7. Let \(\underline{2_ N}\) be a two-element algebra with an \((n+1)\)-ary near-unanimity term function \((n\geq 3)\) and let \(V(\underline{2_ N})\) be the variety generated by [\(\underline{2_ N}\). If there exists a category equivalence \(H\) of \(V(\underline{2_ N})\) then the equivalent category \({\mathcal K}\) is the variety generated by] one of the following algebras: \(A_{\underline{{\mathfrak D}_ n}}\), \(A_{\underline{{\mathfrak D}_ n\cup\{b\}}}\), \(A_{\underline{{\mathfrak D}_ n\cup\rho}}\), \(A_{\underline{{\mathfrak D}_ n\cup\rho\cup\{b\}}}\), where \(\{b\}\) is a one-element subalgebra, \(\underline{{\mathfrak D}_ n}\) is the subalgebra defined in Lemma 4.6, and \(\rho\) is a compatible bounded partial order on \(A\) (or the variety \({\mathcal K}\) is generated by an algebra which is isomorphic to one of these algebras).
Here, \(\underline{{\mathfrak D}_ n}\) is the \(H\)-image of algebra \(\underline{D_ n}\subseteq\underline{2^ n_ N}\) with universe \(D_ n:=\{0,1\}\backslash\{1,\dots,1\}\). The complete formulation of the above-mentioned existence theorem is even more involved and too lengthy to be given here. We refer to the original article.