# Paper «Congruence-based proofs of the recognizability theorems for free many-sorted algebras» published in J. Logic Comput.

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J. Climent Vidal and E. Cosme Llópez.
J. Logic Comput., 30(2):561–633, 2020.

doi:10.1093/logcom/exz032

Abstract

We generalize several recognizability theorems for free single-sorted algebras to free many-sorted algebras and provide, in a uniform way and without using either regular tree grammars or tree automata, purely algebraic proofs of them based on congruences.

Keywords: free many-sorted algebra, recognizability, congruence

# Paper «A characterization of the n-ary many-sorted closure operators and a many-sorted Tarski irredundant basis theorem» in Quaest. Math.

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J. Climent Vidal, E. Cosme Llópez.
A characterization of the \$n\$-ary many-sorted closure operators and a many-sorted Tarski irredundant basis theorem.
Quaest. Math., 42:1427-1444, 2019.

Abstract:

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 ) = Ø, 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.

2020 Mathematics Subject Classification: 06A15, 54A05.

Keywords: S-sorted set, delta of Kronecker, support of an S-sorted set, n-ary many-sorted closure operator, uniform many-sorted closure operator, irredundant basis with respect to a many-sorted closure operator.

# Paper «Eilenberg theorems for many-sorted formations» published in Houston J. Math.

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Juan Climent Vidal, Enric Cosme Llópez.
Eilenberg theorems for many-sorted formations.
Houston J. Math., 45(2):321-369, 2019.

Abstract:

A theorem of Eilenberg establishes that there exists a bijection between the set of all varieties of regular languages and the set of all varieties of finite monoids. In this article after defining, for a fixed set of sorts S and a fixed S-sorted signature Σ, the concepts of formation of congruences with respect to Σ and of formation of Σ-algebras, we prove that the algebraic lattices of all Σ-congruence formations and of all Σ-algebra formations are isomorphic, which is an Eilenberg’s type theorem. Moreover, under a suitable condition on the free Σ-algebras and after defining the concepts of formation of congruences of finite index with respect to Σ, of formation of finite Σ-algebras, and of formation of regular languages with respect to Σ, we prove that the algebraic lattices of all Σ-finite index congruence formations, of all Σ-finite algebra formations, and of all Σ-regular language formations are isomorphic, which is also an Eilenberg’s type theorem.

2020 Mathematics Subject Classification: 08A68, 08A70, 68Q70

Keywords: Many-sorted algebra, support, many-sorted congruence, sat-
uration, cogenerated congruence, many-sorted (finite) algebra formation, many-sorted
(finite index) congruence formation, many-sorted regular language formation.

# Paper «When are profinite many-sorted algebras retracts of ultraproducts of finite many-sorted algebras?» published in Log. J. IGPL

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J. Climent Vidal, E. Cosme Llópez.
When are profinite many-sorted algebras retracts of ultraproducts of finite many-sorted algebras?
Log. J. IGPL, 26(4):381-407, 2018.

Abstract:

For a set of sorts S and an S-sorted signature Σ we prove that a profinite Σ-algebra, i.e. a projective limit of a projective system of finite Σ-algebras, is a retract of an ultraproduct of finite Σ-algebras if the family consisting of the finite Σ-algebras underlying the projective system is with constant support. In addition, we provide a categorial rendering of the above result. Specifically, after obtaining a category where the objects are the pairs formed by a nonempty upward directed preordered set and by an ultrafilter containing the filter of the final sections of it, we show that there exists a functor from the just mentioned category whose object mapping assigns to an object a natural transformation which is a retraction.

2020 Mathematics Subject Classification: 03C20, 08A68, 18A30.

Keywords: support of a many-sorted set, family of many-sorted algebras with constant support, profinite, retract, projective limit, inductive limit, ultraproduct.