Categories and General Algebraic Structures with Applications
https://cgasa.sbu.ac.ir/
Categories and General Algebraic Structures with Applications
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Fri, 01 Jul 2022 00:00:00 +0430
Fri, 01 Jul 2022 00:00:00 +0430

Cover for Vol. 17, No. 1.
https://cgasa.sbu.ac.ir/article_102657.html

Ktheories and Free Inductive Graded Rings in Abstract Quadratic Forms Theories
https://cgasa.sbu.ac.ir/article_101755.html
We build on previous work on multirings ([17]) that providesgeneralizations of the available abstract quadratic forms theories (specialgroups and real semigroups) to the context of multirings ([10], [14]). Herewe raise one step in this generalization, introducing the concept of prespecialhyperfields and expand a fundamental tool in quadratic forms theory to themore general multivalued setting: the Ktheory. We introduce and developthe Ktheory of hyperbolic hyperfields that generalize simultaneously Milnor&rsquo;sKtheory ([11]) and Special Groups Ktheory, developed by DickmannMiraglia ([5]). We develop some properties of this generalized Ktheory, thatcan be seen as a free inductive graded ring, a concept introduced in [2] inorder to provide a solution of Marshall&rsquo;s Signature Conjecture.

Expanding Belnap 2: the dual category in depth
https://cgasa.sbu.ac.ir/article_102443.html
Bilattices, which provide an algebraic tool for simultaneously modelling knowledge and truth, were introduced by N.D. Belnap in a 1977 paper entitled How a computer should think. Prioritised default bilattices include not only Belnap&rsquo;s four values, for &lsquo;true&rsquo; (t), &lsquo;false&rsquo;(f), &lsquo;contradiction&rsquo;(β€) and &lsquo;no information&rsquo; (&perp;), but also indexed families of default values for simultaneously modelling degrees of knowledge and truth. Prioritised default bilattices have applications in a number of areas including artificial intelligence. In our companion paper, we introduced a new family of prioritised default bilattices, Jn, for n β©Ύ 0, with J0 being Belnap&rsquo;s seminal example. We gave a duality for the variety Vn generated by Jn, with the dual category Xn consisting of multisorted topological structures. Here we study the dual category in depth. We axiomatise the category Xn and show that it is isomorphic to a category Yn of singlesorted topological structures. The objects of Yn are ranked Priestley spaces endowed with a continuous retraction. We show how to construct the Priestley dual of the underlying bounded distributive lattice of an algebra in Vn via its dual in Yn; as an application we show that the size of the free algebra FVn(1) is given by a polynomial in n of degree 6.

On some properties of the space of minimal prime ideals of πΆπ (π)
https://cgasa.sbu.ac.ir/article_102622.html
In this article we consider some relations between the topological properties of the spaces X and &nbsp;Min(Cc (X)) with algebraic properties of Cc (X). We observe that the compactness of &nbsp;Min(Cc (X)) is equivalent to the vonNeumann regularity of&nbsp; qc (X), the classical ring of quotients of Cc (X). Furthermore, we show that if π is a strongly zerodimensional space, then each contraction of a minimal prime ideal of πΆ(π) is a minimal prime ideal of Cc(X) and in this case πππ(πΆ(π)) and Min(Cc (X)) are homeomorphic spaces. We also observe that if π is an Fcspace, then&nbsp; Min(Cc (X)) is compact if and only if π is countably basically disconnected if and only if Min(Cc(X)) is homeomorphic with &beta;0X. Finally, by introducing zocideals, countably cozero complemented spaces, we obtain some conditions on X for which &nbsp;Min(Cc (X)) becomes compact, basically disconnected and extremally disconnected.

Universal extensions of specialization semilattices
https://cgasa.sbu.ac.ir/article_102467.html
A specialization semilattice is a join semilattice together with a coarser preorder β satisfying an appropriate compatibility condition. If X is a topological space, then (P(X),&cup;,β) is a specialization semilattice, where x β y if x &sube; Ky, for x, y &sube; X, and K is closure. Specialization semilattices and posets appear as auxiliary structures in many disparate scientific fields, even unrelated to topology. In a former work we showed that every specialization semilattice can be embedded into the specialization semilattice associated to a topological space as above. Here we describe the universal embedding of a specialization semilattice into an additive closure semilattice.

Coverings and liftings of generalized crossed modules
https://cgasa.sbu.ac.ir/article_102491.html
In the theory of crossed modules, considering arbitrary selfactions instead of conjugation allows for the extension of the concept of crossed modules and thus the notion of generalized crossed module emerges. In this paper we give a precise definition for generalized cat1groups and obtain a functor from the category of generalized cat1groups to generalized crossed modules. Further, we introduce the notions of coverings and liftings for generalized crossed modules and investigate properties of these structures. Main objective of this study is to obtain an equivalence between the category of coverings and the category of liftings of a given generalized crossed module (A,B, &alpha;).

On nominal sets with supportpreorder
https://cgasa.sbu.ac.ir/article_102623.html
Each nominal set π can be equipped with a preorder relation βͺ― defined by the notion of support, socalled supportpreorder. This preorder also leads us to the support topology on each nominal set. We study supportpreordered nominal sets and some of their categorical properties in this paper. We also examine the topological properties of support topology, in particular separation axioms.

On injective objects and existence of injective hulls in πTOP/(π, π)
https://cgasa.sbu.ac.ir/article_102640.html
In this paper, motivated by Cagliari and Mantovani, we have obtained a characterization of injective objects (with respect to the class of embeddings in the category πTOP of πtopological spaces) in the comma category πTOP/(π,π), when (π,π) is a stratified πtopological space, with the help of their π0reflection. Further, we have proved that for any πtopological space (π,π), the existence of an injective hull of ((π, π), π ) in the comma category πTOP/(π, π) is equivalent to the existence of an injective hull of its π0reflection ((π Μ,π Μ), π Μ) in the comma category QTOP/(π Μ, π Μ ) (and in the comma category πTOP0/(π Μ, π Μ ), where πTOP0 denotes the category of π0πtopological spaces).

Quantum determinants in ribbon category
https://cgasa.sbu.ac.ir/article_102621.html
The aim of this paper is to introduce an abstract notion of determinant which we call quantum determinant, verifying the properties of the classical one. We introduce R&minus;basis and R&minus;solution on rigid objects of a monoidal π΄π&minus;category, for a compatibility relation R, such that we require the notion of duality introduced by Joyal and Street, the notion given by Yetter and Freyd and the classical one, then we show that R&minus;solutions over a semisimple ribbon π΄π&minus;category form as well a semisimple ribbon π΄π&minus;category. This allows us to define a concept of socalled quantum determinant in ribbon category. Moreover, we establish relations between these and the classical determinants. Some properties of the quantum determinants are exhibited.

Persian Abstracts, Vol. 17, No. 1.
https://cgasa.sbu.ac.ir/article_102658.html

Formal balls of Qcategories
https://cgasa.sbu.ac.ir/article_102914.html
The construction of the formal ball model for metric spaces due to Edalat and Heckmann was generalized to Qcategories by Kostanek and Waszkiewicz, where Q is a commutative and unital quantale. This paper concerns the influence of the structure of the quantale Q on the connection between Yoneda completeness of Qcategories and directed completeness of their sets of formal balls. In the case that Q is the unit interval [0, 1] equipped with a continuous tnorm &amp;, it is shown that in order that Yoneda completeness of each Qcategory be equivalent to directed completeness of its set of formal balls, a necessary and sufficient condition is that the tnorm &amp; is Archimedean.

On saturated prefilter monads
https://cgasa.sbu.ac.ir/article_102934.html
In this paper we show that the prime saturated prefilter monads are supdense and interpolating in saturated prefilter monads. It follows that CNS spaces are the lax algebras for prime saturated prefilter monads. As for the algebraic part, we prove that the EilenbergMoore algebras for saturated prefilter monads are exactly continuous Ilattices.

Action graph of a semigroup act & its functorial connection
https://cgasa.sbu.ac.ir/article_103019.html
In this paper we define Cinduced action graph G(S,a,C;A) corresponding to a semigroup act (S,a,A) and a subset C of S. This generalizes many interesting graphs including Cayley Graph of groups and semigroups, Transformation Graphs (TRAG), Group Action Graphs (GAG), Derangement Action Graphs, Directed Power Graphs of Semigroups etc. We focus on the case when C = S and name the digraph, so obtained, as Action Graph of a Semigroup Act (S, a, A). Some basic structural properties of this graph follow from algebraic properties of the underlying semigroup and its action on the set. Action graph of a strongly faithful act is also studied and graph theoretic characterization of a strongly faithful semigroup act as well as that of idempotents in a semigroup are obtained. We introduce the notion of strongly transitive digraphs and based on this we characterize action graphs of semigroup acts in the class of simple digraphs. The simple fact that morphism between semigroup acts leads to digraph homomorphism between corresponding action graphs, motivates us to represent action graph construction as a functor from the category of semigroup acts to the category of certain digraphs. We capture its functorial properties, some of which signify previous results in terms of Category Theory.

On (semi)topology Lalgebras
https://cgasa.sbu.ac.ir/article_103121.html
Here, we define (semi)topological Lalgebras and some related results are approved. Then we deduce conditions that mention an Lalgebra to be a semitopological or a topological Lalgebra and we check some attributes of them. Chiefly, we display in an Lalgebra L, if (L, β , &tau; ) is a semitopological Lalgebra and {1} is an open set or L is bounded and satisfies the double negation property, then (L,&tau;) is a topological Lalgebra. Finally, we construct a discrete topology on a quotient Lalgebra, under suit able conditions. Also, different kinds of topology such as T0 and Hausdorff are investigated.