Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585313120200701Product preservation and stable units for reflections into idempotent subvarieties1228741410.29252/cgasa.13.1.1ENIsabel A.XarezDepartment of Mathematics, University of Aveiro, Portugal.Joao J.XarezCIDMA - Center for Research and Development in Mathematics and Applications,
Department of Mathematics, University of Aveiro, Portugal.0000-0001-8909-2842Journal Article20190117We give a necessary and sufficient condition for the preservation of finite products by a reflection of a variety of universal algebras into an idempotent subvariety. It is also shown that simple and semi-left-exact reflections into subvarieties of universal algebras are the same. It then follows that a reflection of a variety of universal algebras into an idempotent subvariety has stable units if and only if it is simple and the above-mentioned condition holds.Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585313120200701The non-abelian tensor product of normal crossed submodules of groups23448743710.29252/cgasa.13.1.23ENAlirezaSalemkarFaculty of Mathematical Sciences, Shahid Beheshti University, Tehran 19839, Iran.TaherehFakhr TahaFaculty of Mathematical Sciences, Shahid Beheshti University, Tehran 19839, Iran.Journal Article20190325In this article, the notions of non-abelian tensor and exterior products of two normal crossed submodules of a given crossed module of groups are introduced and some of their basic properties are established. In particular, we investigate some common properties between normal crossed modules and their tensor products, and present some bounds on the nilpotency class and solvability length of the tensor product, provided such information is given at least on one of the normal crossed submodules.Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585313120200701Distributive lattices with strong endomorphism kernel property as direct sums45548751210.29252/cgasa.13.1.45ENJaroslavGuricanDepartment of Algebra and Geometry,
Faculty of Mathematics, Physics and Informatics, Comenius University Bratislava, Slovakia.0000-0002-2857-161XJournal Article20200607Unbounded distributive lattices which have strong endomorphism kernel property (SEKP) introduced by Blyth and Silva in [3] were fully characterized in [11] using Priestley duality (see TheoremÂ 2.8}). We shall determine the structure of special elements (which are introduced afterÂ Theorem 2.8 under the name strong elements) and show that these lattices can be considered as a direct product of three lattices, a lattice with exactly one strong element, a lattice which is a direct sum of 2 element lattices with distinguished elements 1 and a lattice which is a direct sum of 2 element lattices with distinguished elements 0, and the sublattice of strong elements is isomorphic to a product of last two mentioned lattices.Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585313120200701Separated finitely supported $Cb$-sets55828741310.29252/cgasa.13.1.55ENKhadijehKeshvardoostDepartment of Mathematics, Velayat University, Iranshahr, Sistan and Baluchestan, Iran.MojganMahmoudiDepartment of Mathematics,
Shahid Beheshti University, Tehran 19839, Iran.Journal Article20190405The monoid $Cb$ of name substitutions and the notion of finitely supported $Cb$-sets introduced by Pitts as a generalization of nominal sets. A simple finitely supported $Cb$-set is a one point extension of a cyclic nominal set. The support map of a simple finitely supported $Cb$-set is an injective map. Also, for every two distinct elements of a simple finitely supported $Cb$-set, there exists an element of the monoid $Cb$ which separates them by making just one of them into an element with the empty support.<br />In this paper, we generalize these properties of simple finitely supported $Cb$-sets by modifying slightly the notion of the support map; defining the notion of $mathsf{2}$-equivariant support map; and introducing the notions of s-separated and z-separated finitely supported $Cb$-sets. We show that the notions of s-separated and z-separated coincide for a finitely supported $Cb$-set whose support map is $mathsf{2}$-equivariant. Among other results, we find a characterization of simple s-separated (or z-separated) finitely supported $Cb$-sets. Finally, we show that some subcategories of finitely supported $Cb$-sets with injective equivariant maps which constructed applying the defined notions are reflective.Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585313120200701A classification of hull operators in archimedean lattice-ordered groups with unit831048755210.29252/cgasa.13.1.83ENRicardo E.CarreraDepartment of Mathematics, Nova Southeastern University, 3301 College Ave., Fort Lauderdale, FL, 33314, USA.Anthony W.HagerDepartment of Mathematics and CS, Wesleyan University, Middletown, CT 06459.Journal Article20191026The category, or class of algebras, in the title is denoted by $bf W$. A hull operator (ho) in $bf W$ is a reflection in the category consisting of $bf W$ objects with only essential embeddings as morphisms. The proper class of all of these is $bf hoW$. The bounded monocoreflection in $bf W$ is denoted $B$. We classify the ho's by their interaction with $B$ as follows. A ``word'' is a function $w: {bf hoW} longrightarrow {bf W}^{bf W}$ obtained as a finite composition of $B$ and $x$ a variable ranging in $bf hoW$. The set of these,``Word'', is in a natural way a partially ordered semigroup of size $6$, order isomorphic to ${rm F}(2)$, the free $0-1$ distributive lattice on $2$ generators. Then, $bf hoW$ is partitioned into $6$ disjoint pieces, by equations and inequations in words, and each piece is represented by a characteristic order-preserving quotient of Word ($approx {rm F}(2)$). Of the $6$: $1$ is of size $geq 2$, $1$ is at least infinite, $2$ are each proper classes, and of these $4$, all quotients are chains; another $1$ is a proper class with unknown quotients; the remaining $1$ is not known to be nonempty and its quotients would not be chains.Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585313120200701The symmetric monoidal closed category of cpo $M$-sets1051248743410.29252/cgasa.13.1.105ENHalimehMoghbeliDepartment of Mathematics, Faculty of Science, University of Jiroft, Jiroft, Iran0000-0001-7316-4565Journal Article20190329In this paper, we show that the category of directed complete posets with bottom elements (cpos) endowed with an action of a monoid $M$ on them forms a monoidal category. It is also proved that this category is symmetric closed.Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585313120200701Crossed squares, crossed modules over groupoids and cat$^{bf {1-2}}-$groupoids1251428751110.29252/cgasa.13.1.125ENSedatTemelDepartment of Mathematics, Faculty of Arts and Science, Recep Tayyip Erdogan University, Rize, Turkey.0000-0001-6553-8758Journal Article20200607The aim of this paper is to introduce the notion of cat$^{bf {1}}-$groupoids which are the groupoid version of cat$^{bf {1}}-$groups and to prove the categorical equivalence between crossed modules over groupoids and cat$^{bf {1}}-$groupoids. In section 4 we introduce the notions of crossed squares over groupoids and of cat$^{bf {2}}-$groupoids, and then we show their categories are equivalent. These equivalences enable us to obtain more examples of groupoids.Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585313120200701Tense like equality algebras1431668746510.29252/cgasa.13.1.143ENMohammad AliHashemiDepartment of Mathematics, Payame Noor University, P.O.Box 19395-3697, Tehran, Iran.RajabaliBorzooeiDepartment of Mathematics, Shahid Beheshti University, Tehran, Iran.Journal Article20190423In this paper, first we define the notion of involutive operator on bounded involutive equality algebras and by using it, we introduce a new class of equality algebras that we called it a tense like equality algebra. Then we investigate some properties of tense like equality algebra. For two involutive bounded equality algebras and an equality homomorphism between them, we prove that the tense like equality algebra structure can be transfer by this equality homomorphism. Specially, by using a bounded involutive equality algebra and quotient structure of it, we construct a quotient tense like equality algebra. Finally, we investigate the relation between tense like equality algebras and tense MV-algebras.