La toponímia com a eina d'aproximació al romanç andalusí: el cas de Mallorca i Menorca

Després de fer un repàs a les obres que s’han ocupat del tema, s’intenta, a partir de la toponímia conservada en la documentació i dels noms de lloc subsistents, classificar els fenòmens fonètics que semblen caracteritzar el parlar romànic anterior a la conquista catalana en el marc geogràfic de les Balears estrictes

K4-free graphs as a free algebra

Graphs of treewidth at most two are the ones excluding the clique with four vertices (K4) as a minor, or equivalently, the graphs whose biconnected components are series-parallel. We turn those graphs into a finitely presented free algebra,answering positively a question by Courcelle and Engelfriet, in the case of treewidth two. First we propose a syntax for denoting these graphs: in addition to parallel composition and series composition, it suffices to consider the neutral elements of those operations and a unary transpose operation. Then we give a finite equationa lpresentation and we prove it complete: two terms from the syntax are congruent if and only if they denote the same graph

When are profinite many-sorted algebras retracts of ultraproducts of finite many-sorted algebras?

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

Eilenberg Theorems for Many-sorted Formations

A theorem of Eilenberg establishes that there exists a bijectionbetween the set of all varieties of regular languages and the set of all vari-eties of finite monoids. In this article after defining, for a fixed set of sortsSand a fixedS-sorted signature Σ, the concepts of formation of congruenceswith respect to Σ and of formation of Σ-algebras, we prove that the alge-braic lattices of all Σ-congruence formations and of all Σ-algebra formationsare isomorphic, which is an Eilenberg's type theorem. Moreover, under asuitable condition on the free Σ-algebras and after defining the concepts offormation of congruences of finite index with respect to Σ, of formation offinite Σ-algebras, and of formation of regular languages with respect to Σ, weprove that the algebraic lattices of all Σ-finite index congruence formations,of all Σ-finite algebra formations, and of all Σ-regular language formationsare isomorphic, which is also an Eilenberg's type theorem

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 ) = 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

Functoriality of the Schmidt construction

After proving, in a purely categorial way, that the inclusion functor InAlg(Σ) from Alg(Σ)⁠, the category of many-sorted Σ-algebras, to PAlg(Σ)⁠, the category of many-sorted partial Σ-algebras, has a left adjoint FΣ⁠, the (absolutely) free completion functor, we recall, in connection with the functor FΣ⁠, the generalized recursion theorem of Schmidt, which we will also call the Schmidt construction. Next, we define a category Cmpl(Σ)⁠, of Σ-completions, and prove that FΣ⁠, labelled with its domain category and the unit of the adjunction of which it is a part, is a weakly initial object in it. Following this, we associate to an ordered pair (α,f)⁠, where α=(K,γ,α) is a morphism of Σ-completions from F=(C,F,η) to G=(D,G,ρ) and f a homomorphism of D from the partial Σ-algebra A to the partial Σ-algebra B⁠, a homomorphism Υ G,0α(f):Schα(f)⟶B⁠. We then prove that there exists an endofunctor, Υ G,0α, of Mortw(D)⁠, the twisted morphism category of D⁠, thus showing the naturalness of the previous construction. Afterwards, we prove that, for every Σ-completion G=(D,G,ρ)⁠, there exists a functor ΥG from the comma category (Cmpl(Σ)↓G) to End(Mortw(D))⁠, the category of endofunctors of Mortw(D)⁠, such that ΥG,0⁠, the object mapping of ΥG⁠, sends a morphism of Σ-completion of Cmpl(Σ) with codomain G⁠, to the endofunctor ΥG,0