39,759 research outputs found

    Lattice of closure endomorphisms of a Hilbert algebra

    Full text link
    A closure endomorphism of a Hilbert algebra A is a mapping that is simultaneously an endomorphism of and a closure operator on A. It is known that the set CE of all closure endomorphisms of A is a distributive lattice where the meet of two elements is defined pointwise and their join is given by their composition. This lattice is shown in the paper to be isomorphic to the lattice of certain filters of A, anti-isomorphic to the lattice of certain closure retracts of A, and compactly generated. The set of compact elements of CE coincides with the adjoint semilattice of A, conditions under which two Hilbert algebras have isomorphic adjoint semilattices (equivalently, minimal Brouwerian extensions) are discussed. Several consequences are drawn also for implication algebras.Comment: 16 pages, no figures, submitted to Algebra Universalis (under review since 24.11.2015

    Semisimple Varieties of Implication Zroupoids

    Get PDF
    It is a well known fact that Boolean algebras can be defined using only implication and a constant. In 2012, this result was extended to De Morgan algebras in [8] which led Sankappanavar to introduce, and investigate, the variety I of implication zroupoids generalizing De Morgan algebras. His investigations were continued in [3] and [4] in which several new subvarieties of I were introduced and their relationships with each other and with the varieties of [8] were explored. The present paper is a continuation of [8] and [3]. The main purpose of this paper is to determine the simple algebras in I. It is shown that there are exactly five simple algebras in I. From this description we deduce that the semisimple subvarieties of I are precisely the subvarieties of the variety generated by these 5 simple I-zroupoids and are locally finite. It also follows that the lattice of semisimple subvarieties of I is isomorphic to the direct product of a 4-element Boolean lattice and a 4-element chain.Comment: 21 page

    Direct sums and products in topological groups and vector spaces

    Full text link
    We call a subset AA of an abelian topological group GG: (i) absolutelyabsolutely CauchyCauchy summablesummable provided that for every open neighbourhood UU of 00 one can find a finite set F⊆AF\subseteq A such that the subgroup generated by A∖FA\setminus F is contained in UU; (ii) absolutelyabsolutely summablesummable if, for every family {za:a∈A}\{z_a:a\in A\} of integer numbers, there exists g∈Gg\in G such that the net \left\{\sum_{a\in F} z_a a: F\subseteq A\mbox{ is finite}\right\} converges to gg; (iii) topologicallytopologically independentindependent provided that 0∈̞A0\not \in A and for every neighbourhood WW of 00 there exists a neighbourhood VV of 00 such that, for every finite set F⊆AF\subseteq A and each set {za:a∈F}\{z_a:a\in F\} of integers, ∑a∈Fzaa∈V\sum_{a\in F}z_aa\in V implies that zaa∈Wz_aa\in W for all a∈Fa\in F. We prove that: (1) an abelian topological group contains a direct product (direct sum) of Îș\kappa-many non-trivial topological groups if and only if it contains a topologically independent, absolutely (Cauchy) summable subset of cardinality Îș\kappa; (2) a topological vector space contains R(N)\mathbb{R}^{(\mathbb{N})} as its subspace if and only if it has an infinite absolutely Cauchy summable set; (3) a topological vector space contains RN\mathbb{R}^{\mathbb{N}} as its subspace if and only if it has an R(N)\mathbb{R}^{(\mathbb{N})} multiplier convergent series of non-zero elements. We answer a question of Hu\v{s}ek and generalize results by Bessaga-Pelczynski-Rolewicz, Dominguez-Tarieladze and Lipecki

    The relation of semiadjacency of ∩\cap-semigroups of transformations

    Full text link
    We consider two relations on a ∩\cap-semigroup of partial functions of a given set: the inclusion of domains and the semiadjacencity (i.e., the inclusion of the image of the first function into the domain of the second), which characterized with an abstract point of view using the elementary system of axioms, i.e., system conditions, recorded in the language narrow predicate calculus with equality

    Graph quasivarieties

    Full text link
    Introduced by C. R. Shallon in 1979, graph algebras establish a useful connection between graph theory and universal algebra. This makes it possible to investigate graph varieties and graph quasivarieties, i.e., classes of graphs described by identities or quasi-identities. In this paper, graph quasivarieties are characterized as classes of graphs closed under directed unions of isomorphic copies of finite strong pointed subproducts.Comment: 15 page
    • 

    corecore