Categories and General Algebraic Structures with Applications
https://cgasa.sbu.ac.ir/
Categories and General Algebraic Structures with Applicationsendaily1Tue, 01 Aug 2023 00:00:00 +0330Tue, 01 Aug 2023 00:00:00 +0330Cover for Vol. 19, No. 1.
https://cgasa.sbu.ac.ir/article_103880.html
Cancel culture
https://cgasa.sbu.ac.ir/article_103580.html
Let A, B, C, and D be posets. Assume C and D are finite with a greatest element. Also assume that AC&nbsp;&cong;B D. Then there exist posets E, X, Y , and Z such that A &cong;E X, B &cong;E Y&nbsp;, C&cong;Y &times;Z, and D&cong;X&times;Z. If C&cong;D, then A&cong;B. This generalizes a theorem of J&oacute;nsson and McKenzie, who proved it when A and B were meet-semilattices.Composition series on (Rees) congruences of S-acts.
https://cgasa.sbu.ac.ir/article_103375.html
In this paper, we study composition series of subacts or congruences of S-acts. It is shown that composition series of subacts are exactly those that are both Rees artinian and Rees noetherian, i.e. those satisfying both ascending and descending chain conditions on subacts. But this is not valid for the case of composition series of congruences in general. We prove that the properties of having composition series of subacts or congruences are inherited in Rees short exact sequences. Also, we discuss whenever two composition series of subacts or congruences have the same length and they are equivalent.Determinant and rank functions in semisimple pivotal Ab-categories
https://cgasa.sbu.ac.ir/article_103829.html
We investigate and generalize quantum determinants to semisimple spherical and pivotal categories. It is well known that traces are preserved by strong tensor functors; we show on one hand that in fact, weaker conditions on a functor are sufficient to continue preserving traces. On the other hand, we prove that these determinants are well-behaved under strong tensor functors. Further, we introduce a notion of domination rank for objects of a semisimple pivotal category and prove similar properties of the ordinary case. Furthermore, we expand the determinantal and McCoy ranks to introduce a morphism quantum rank function on a semisimple pivotal category.On free acts over semigroups and their lattices of radical subacts
https://cgasa.sbu.ac.ir/article_103736.html
This study aims to investigate free objects in the category of acts over an arbitrary semigroup S. We consider two generalizations of free acts over arbitrary semigroups, namely acts with conditions (F1) and (F2), and give some new results about (minimal) prime subacts and radical subacts of any S-act with condition (F1). Furthermore, some lattice structures for some collections of radical subacts of free S-acts are introduced. We also obtain some results about the relationship between radical subacts of free S-acts and radical ideals of S. Moreover, for any prime ideal P of a semigroup S with a zero, we find a one-to-one correspondence between the collections of P-prime subacts of any two free S-acts. Also, it is shown that all free S-acts have isomorphic lattices of radical multiplication subacts.Morita equivalence of certain crossed products
https://cgasa.sbu.ac.ir/article_103842.html
We introduce an alternative criterion for Morita equivalence over graded tensor categories using equivariant centers and equivariantizations. While Morita equivalence has been extensively studied in the context of fusion categories, primarily through the examination of their centers, recent advancements have broadened its scope to encompass graded tensor categories. This paper presents a novel criterion for characterizing Morita equivalence in graded tensor categories by leveraging equivariant centers and equivariantizations. Notably, the identification of Morita equivalence can be expedited when the Brauer Picard groups are known, offering an efficient approach to establishing the equivalence relationship.To generalize the properties of fusion categories to finite tensor categories, we utilize theconcept of an exact module category, which was introduced by Etingof and Ostrik. Exact module categories offer an intermediary restriction between the semisimplemodule categories of a fusion category and more general cases that may not be semisimpleor finite.On C-injective generalized hyper S-acts
https://cgasa.sbu.ac.ir/article_103881.html
This paper explores generalized hyper S-acts (GHS-acts) over a hypermonoid S as generalizations of monoid acts within the context of algebraic hyperstructures. Specifically, we extend the definition of C-injectivity to GHS-acts and investigate their internal and homological properties. It is established that for being GHS-injectivity of GHS-acts with a fixed element, it suffices to consider allinclusions from cyclic GHS-subacts into indecomposable ones. Then we introducenew concepts known as semi-injectivity and semi-C-injectivity. By providing examples, we demonstrate that injectivity and semi-injectivity (C-injectivity and semiC-injectivity) are different concepts for GHS-acts, whereas they are the same in the context of acts over monoids. It is also shown that all pure GHS-acts are injective if and only if all pure cyclic GHS-acts are C-injective. Furthermore, we establish an equivalent condition on a hypermonoid S such that all quotients of SS exhibit semi-injectivity. Finally, we derive an equivalent condition for a hypermonoid to beclassified as semi-injective.Green's relations on ordered n-ary semihypergroups
https://cgasa.sbu.ac.ir/article_103858.html
In this paper, we introduce the concept of weak $i$-hyperfilters of ordered $n$-arysemihypergroups, where a positive integer 1 &le; i &le; n and n &ge; 2, and then discuss theirrelated properties. We define the Green&rsquo;s relations $M_i, J , H$ and $K$ on ordered $n$-arysemihypergroups and investigate the relationships between the Green&rsquo;s relations and theequivalence relation $W_i$, which is generated by the weak $i$-hyperfilters. Also, we givethe characterizations of intra-regular ordered $n$-ary semihypergroups via the propertiesof weak $i$-hyperfilters. Finally, we introduce the concepts of $(i-, &Lambda;-)$duo ordered $n$-arysemihypergroups and establish some interesting properties.Persian Abstracts, Vol. 19, No. 1.
https://cgasa.sbu.ac.ir/article_103882.html
Idempotent 2x2 matrices over linearly ordered abelian groups
https://cgasa.sbu.ac.ir/article_104001.html
In this paper we study multiplicative semigroups of $2\times 2$ matrices over a linearly ordered abelian group with an externally added bottom element. The multiplication of such a semigroup is defined by replacing addition and multiplication by join and addition in the usual formula defining matrix multiplication. We show that there are four types of idempotents in this semigroup and we determine which of them are $0$-primitive. We also prove that the poset of idempotents with respect to the natural order is a lattice. It turns out that this matrix semigroup is inverse or orthodox if and only if the abelian group is trivial.