The dual equivalence of equations and coequations for automata

Because of the isomorphism (X x A) -> X = X -> (A -> X), the transition structure t: X -> (A -> X) 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) tha

Regular Varieties of Automata and Coequations

In this paper we use a duality result between equations and coequations for automata, proved by Ballester-Bolinches, Cosme-Ll´opez, 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 dual equivalence of equations and coequations for automata

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A characterization of the n-ary many-sorted closure operators and a many-sorted Tarski irredundant basis theorem

A theorem of single-sorted algebra states that, for a closure space (A, J) and a natural number n, the closure operator J on the set A is n-ary if and only if there exists a single-sorted signature Σ and a Σ-algebra A such that every operation of A is of an arity ≤ n and J = SgA, where SgA is the subalgebra generating operator on A determined by A. On the other hand, a theorem of Tarski asserts that if J is an n-ary closure operator on a set A with n ≥ 2, then, for every i, j ∈ IrB(A, J), where IrB(A, J) is the set of all natural numbers which have the property of being the cardinality of an irredundant basis (≡ minimal generating set) of A with respect to J, if i < j and {i+ 1, . . . , j −1} ∩IrB(A, J) = ∅, then j −i ≤ n−1. In this article we state and prove the many-sorted counterparts of the above theorems. But, we remark, regarding the first one under an additional condition: the uniformity of the many-sorted closure operator.  Key words: S-sorted set, delta of Kronecker, support of an S-sorted set, n-ary manysorted closure operator, uniform many-sorted closure operator, irredundant basis with respect to a many-sorted closure operator