Locally finite M-solid varieties of semigroups

An algebra of type τ is said to be locally finite if all its finitely generated subalgebras are finite. A class K of algebras of type τ is called locally finite if all its elements are locally finite. It is well-known (see [2]) that a variety of algebras of the same type τ is locally finite iff all its finitely generated free algebras are finite. A variety V is finitely based if it admits a finite basis of identities, i.e. if there is a finite set σ of identities such that V = ModΣ, the class of all algebras of type τ which satisfy all identities from Σ. Every variety which is generated by a finite algebra is locally finite. But there are finite algebras which are not finitely based. For semigroup varieties, Perkins proved that the variety generated by the five-element Brandt-semigroup ${\displaystyle B¹₂=\{beg\in \{\pm atrix\}0\&0\backslash 0\&0end\{\pm atrix\},beg\in \{\pm atrix\}1\&0\backslash 0\&0end\{\pm atrix\},beg\in \{\pm atrix\}0\&1\backslash 0\&0end\{\pm atrix\},beg\in \{\pm atrix\}0\&0\backslash 1\&0end\{\pm atrix\},beg\in \{\pm atrix\}0\&0\backslash 0\&1end\{\pm atrix\}\}}$$$\left(\begin{array}{cc}0& 0\\ 0& 0\end{array}\right)$$, $$\left(\begin{array}{cc}1& 0\\ 0& 0\end{array}\right)$$, $$\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)$$, $$\left(\begin{array}{cc}0& 0\\ 1& 0\end{array}\right)$$, $$\left(\begin{array}{cc}0& 0\\ 0& 1\end{array}\right)$$}!$! is not finitely based ([9], [10]). An identity s ≈ t is called a hyperidentity of a variety V if whenever the operation symbols occurring in s and in t, respectively, are replaced by any terms of V of the appropriate arity, the identity which results, holds in V. A variety V is called solid if every identity of V also holds as a hyperidentity in V. If we apply only substitutions from a set M we speak of M-hyperidentities and M-solid varieties. In this paper we use the theory of M-solid varieties to prove that a type (2) M-solid variety of the form ${\displaystyle V=H_{M}Mod\{F(x₁,F(x₂,x₃))\approx F(F(x₁,x₂),x₃)\}}$, which consists precisely of all algebras which satisfy the associative law as an M -hyperidentity is locally finite iff the hypersubstitution which maps F to the word x₁x₂x₁ or to the word x₂x₁x₂ belongs to M and that V is finitely based if it is locally finite.