### A band generated by two semilattices is regular.

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We prove here an Eilenberg type theorem: the so-called conjunctive varieties of rational languages correspond to the pseudovarieties of finite semilattice-ordered monoids. Taking complements of members of a conjunctive variety of languages we get a so-called disjunctive variety. We present here a non-trivial example of such a variety together with an equational characterization of the corresponding pseudovariety.

In a previous paper, the authors studied the polynomial closure of a variety of languages and gave an algebraic counterpart, in terms of Mal’cev products, of this operation. They also formulated a conjecture about the algebraic counterpart of the boolean closure of the polynomial closure – this operation corresponds to passing to the upper level in any concatenation hierarchy. Although this conjecture is probably true in some particular cases, we give a counterexample in the general case. Another...

In a previous paper, the authors studied the polynomial closure of a variety of languages and gave an algebraic counterpart, in terms of Mal'cev products, of this operation. They also formulated a conjecture about the algebraic counterpart of the boolean closure of the polynomial closure – this operation corresponds to passing to the upper level in any concatenation hierarchy. Although this conjecture is probably true in some particular cases, we give a counterexample in the general case....

We define semantically a partial multiplication on the lattice of all e–varieties of regular semigroups. In the case that the first factor is an e–variety of orthodox semigroups we describe our multiplication syntactically in terms of biinvariant congruences.

A multiplication of e-varieties of regular $E$-solid semigroups by inverse semigroup varieties is described both semantically and syntactically. The associativity of the multiplication is also proved.

It is studied how taking the inverse image by a sliding block code affects the syntactic semigroup of a sofic subshift. The main tool are ζ-semigroups, considered as recognition structures for sofic subshifts. A new algebraic invariant is obtained for weak equivalence of sofic subshifts, by determining which classes of sofic subshifts naturally defined by pseudovarieties of finite semigroups are closed under weak equivalence. Among such classes are the classes of almost finite type subshifts...

We show in an additive inverse regular semiring $(S,+,\xb7)$ with ${E}^{\u2022}\left(S\right)$ as the set of all multiplicative idempotents and ${E}^{+}\left(S\right)$ as the set of all additive idempotents, the following conditions are equivalent: (i) For all $e,f\in {E}^{\u2022}\left(S\right)$, $ef\in {E}^{+}\left(S\right)$ implies $fe\in {E}^{+}\left(S\right)$. (ii) $(S,\xb7)$ is orthodox. (iii) $(S,\xb7)$ is a semilattice of groups. This result generalizes the corresponding result of regular ring.

Completely regular semigroups are unions of their (maximal) subgroups with the unary operation within their maximal subgroups. As such they form a variety whose lattice of subvarieties is denoted by $\mathcal{L}\left(\mathcal{C}\mathcal{R}\right)$. We construct a 60-element $\cap $-subsemilattice and a 38-element sublattice of $\mathcal{L}\left(\mathcal{C}\mathcal{R}\right)$. The bulk of the paper consists in establishing the necessary joins for which it uses Polák’s theorem.

In Universal Algebra, identities are used to classify algebras into collections, called varieties and hyperidentities are use to classify varieties into collections, called hypervarities. The concept of a hypersubstitution is a tool to study hyperidentities and hypervarieties. Generalized hypersubstitutions and strong identities generalize the concepts of a hypersubstitution and of a hyperidentity, respectively. The set of all generalized hypersubstitutions forms a monoid. In...