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Lattice of closure endomorphisms of a Hilbert algebra
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
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
We call a subset of an abelian topological group : (i)
provided that for every open neighbourhood of one
can find a finite set such that the subgroup generated by
is contained in ; (ii) if, for every
family of integer numbers, there exists such that the
net \left\{\sum_{a\in F} z_a a: F\subseteq A\mbox{ is finite}\right\}
converges to ; (iii) provided that and for every neighbourhood of there exists a neighbourhood of
such that, for every finite set and each set of integers, implies that for all
. We prove that: (1) an abelian topological group contains a direct
product (direct sum) of -many non-trivial topological groups if and
only if it contains a topologically independent, absolutely (Cauchy) summable
subset of cardinality ; (2) a topological vector space contains
as its subspace if and only if it has an infinite
absolutely Cauchy summable set; (3) a topological vector space contains
as its subspace if and only if it has an
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 -semigroups of transformations
We consider two relations on a -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
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
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