Because of the isomorphism (X × A) → X ∼ = X → X A , the transition structure α : X → X A of a deterministic automaton with state set X and with inputs from an alphabet A can be viewed both as an algebra and as a coalgebra. Here we will use this algebra-coalgebra duality of automata as a common perspective for the study of equations and coequations. Equations are sets of pairs of words (v, w) that are satisfied by a state x ∈ X if they lead to the same state: x v = x w . Dually, coequations are sets of languages and are satisfied by x if the language accepted by x belongs to that set. For every automaton (X, α), we define two new automata: free(X, α) and cofree(X, α) that represent, respectively, the greatest set of equations and the smallest set of coequations satisfied by (X, α). Both constructions are shown to be functorial, that is, they act also on automaton homomorphisms. The automaton free(X, α) is isomorphic to the socalled transition monoid of (X, α), and thereby, cofree(X, α) can be seen as its dual. Our main result is that the restrictions of free and cofree to, respectively, preformations of languages and to quotients A * /C of A * with respect to a congruence relation C, form a dual equivalence. In the present context, preformations of languages are sets of -not necessarily regular -languages that are complete atomic Boolean algebras closed under left and right language derivatives. This result is used to give an alternative definition of the notion of "varieties of regular languages" introduced by Eilenberg. This definition, based on equations and coequations, underscores the prominent role of congruences in this kind of results. As a consequence, we present a variant of Eilenberg's celebrated variety theorem for varieties of monoids (in the sense of Birkhoff) and varieties of languages. AcknowledgementsWe are much obliged to the two anonymous referees: their comments and suggestions have greatly improved our paper.

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Abstract. In this paper we use a duality result between equations and coequations for automata, proved by Ballester-Bolinches, Cosme-Llópez, and Rutten to characterize nonempty classes of deterministic automata that are closed under products, subautomata, homomorphic images, and sums. One characterization is as classes of automata defined by regular equations and the second one is as classes of automata satisfying sets of coequations called varieties of languages. We show how our results are related to Birkhoff's theorem for regular varieties.

The main goal in this paper is to use a dual equivalence in automata theory started in [25] and developed in [3] to prove a general version of the Eilenberg-type theorem presented in [4]. Our principal results confirm the existence of a bijective correspondence between three concepts; formations of monoids, formations of languages and formations of congruences. The result does not require finiteness on monoids, nor regularity on languages nor finite index conditions on congruences. We relate our work to other results in the field and we include applications to non-r-disjunctive languages, Reiterman's equational description of pseudovarieties and varieties of monoids.

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In this paper we describe some algorithms to identify permutable and Sylow-permutable subgroups of finite groups, Dedekind and Iwasawa finite groups, and finite T-groups (groups in which normality is transitive), PT-groups (groups in which permutability is transitive), and PST-groups (groups in which Sylow permutability is transitive). These algorithms have been implemented in a package for the computer algebra system GAP. MSC:20D10, 20D20, 20-04

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The aim of this paper is to present some contributions to the theory of finite transformation monoids. The dominating influence that permutation groups have on transformation monoids is used to describe and characterise transitive transformation monoids and primitive transitive transformation monoids. We develop a theory that not only includes the analogs of several important theorems of the classical theory of permutation groups but also contains substantial information about the algebraic structure of the transformation monoids. Open questions naturally arising from the substantial paper of Steinberg [A theory of transformation monoids: combinatorics and representation theory. Electron. J. Combin. 17 (2010), no. 1, Research Paper 164, 56 pp] have been answered. Our results can also be considered as a further development in the hunt for a solution of theČerný conjecture.