Pub Date : 2022-07-01DOI: 10.52547/cgasa.2022.102621
H. Choulli, Khalid Draoui, H. Mouanis
. 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− basis and R− solution on rigid objects of a monoidal 𝐴𝑏 − 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− solutions over a semisimple ribbon 𝐴𝑏 − category form as well a semisimple ribbon 𝐴𝑏 − category. This allows us to define a concept of so-called quantum determinant in ribbon category. Moreover, we establish relations between these and the classical determinants. Some properties of the quantum determinants are exhibited.
{"title":"Quantum determinants in ribbon category","authors":"H. Choulli, Khalid Draoui, H. Mouanis","doi":"10.52547/cgasa.2022.102621","DOIUrl":"https://doi.org/10.52547/cgasa.2022.102621","url":null,"abstract":". 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− basis and R− solution on rigid objects of a monoidal 𝐴𝑏 − 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− solutions over a semisimple ribbon 𝐴𝑏 − category form as well a semisimple ribbon 𝐴𝑏 − category. This allows us to define a concept of so-called quantum determinant in ribbon category. Moreover, we establish relations between these and the classical determinants. Some properties of the quantum determinants are exhibited.","PeriodicalId":41919,"journal":{"name":"Categories and General Algebraic Structures with Applications","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47722746","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-07-01DOI: 10.52547/cgasa.2022.102623
Aliyeh Hossinabadi, M. Haddadi, Khadijeh Keshvardoost
. Each nominal set 𝑋 can be equipped with a preorder relation ⪯ defined by the notion of support, so-called support-preorder. This preorder also leads us to the support topology on each nominal set. We study support-preordered nominal sets and some of their categorical properties in this paper. We also examine the topological properties of support topology, in particular separation axioms.
{"title":"On nominal sets with support-preorder","authors":"Aliyeh Hossinabadi, M. Haddadi, Khadijeh Keshvardoost","doi":"10.52547/cgasa.2022.102623","DOIUrl":"https://doi.org/10.52547/cgasa.2022.102623","url":null,"abstract":". Each nominal set 𝑋 can be equipped with a preorder relation ⪯ defined by the notion of support, so-called support-preorder. This preorder also leads us to the support topology on each nominal set. We study support-preordered nominal sets and some of their categorical properties in this paper. We also examine the topological properties of support topology, in particular separation axioms.","PeriodicalId":41919,"journal":{"name":"Categories and General Algebraic Structures with Applications","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48074330","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-07-01DOI: 10.52547/cgasa.2021.101755
K. Roberto, H. Mariano
. We build on previous work on multirings ( [17]) that provides generalizations of the available abstract quadratic forms theories (special groups and real semigroups) to the context of multirings ( [10], [14]). Here we raise one step in this generalization, introducing the concept of pre-special hyperfields and expand a fundamental tool in quadratic forms theory to the more general multivalued setting: the K-theory. We introduce and develop the K-theory of hyperbolic hyperfields that generalize simultaneously Milnor’s K-theory ( [11]) and Special Groups K-theory, developed by Dickmann-Miraglia ( [5]). We develop some properties of this generalized K-theory, that can be seen as a free inductive graded ring, a concept introduced in [2] in order to provide a solution of Marshall’s Signature Conjecture.
{"title":"K-theories and Free Inductive Graded Rings in Abstract Quadratic Forms Theories","authors":"K. Roberto, H. Mariano","doi":"10.52547/cgasa.2021.101755","DOIUrl":"https://doi.org/10.52547/cgasa.2021.101755","url":null,"abstract":". We build on previous work on multirings ( [17]) that provides generalizations of the available abstract quadratic forms theories (special groups and real semigroups) to the context of multirings ( [10], [14]). Here we raise one step in this generalization, introducing the concept of pre-special hyperfields and expand a fundamental tool in quadratic forms theory to the more general multivalued setting: the K-theory. We introduce and develop the K-theory of hyperbolic hyperfields that generalize simultaneously Milnor’s K-theory ( [11]) and Special Groups K-theory, developed by Dickmann-Miraglia ( [5]). We develop some properties of this generalized K-theory, that can be seen as a free inductive graded ring, a concept introduced in [2] in order to provide a solution of Marshall’s Signature Conjecture.","PeriodicalId":41919,"journal":{"name":"Categories and General Algebraic Structures with Applications","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42706558","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-22DOI: 10.52547/cgasa.2022.102467
P. Lipparini
. A specialization semilattice is a join semilattice together with a coarser preorder ⊑ satisfying an appropriate compatibility condition. If X is a topological space, then ( P ( X ) , ∪ , ⊑ ) is a specialization semilattice, where x ⊑ y if x ⊆ Ky , for x, y ⊆ X , and K is closure. Specialization semilattices and posets appear as auxiliary structures in many disparate scientific fields, even unrelated to topology. For short, the notion is useful since it allows us to consider a relation of “being generated by” with no need to require the existence of an actual “closure” or “ hull”, which is problematic in certain contexts. 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. We notice that a categorical argument guarantees the existence of universal embeddings in many parallel situations.
{"title":"Universal extensions of specialization semilattices","authors":"P. Lipparini","doi":"10.52547/cgasa.2022.102467","DOIUrl":"https://doi.org/10.52547/cgasa.2022.102467","url":null,"abstract":". A specialization semilattice is a join semilattice together with a coarser preorder ⊑ satisfying an appropriate compatibility condition. If X is a topological space, then ( P ( X ) , ∪ , ⊑ ) is a specialization semilattice, where x ⊑ y if x ⊆ Ky , for x, y ⊆ X , and K is closure. Specialization semilattices and posets appear as auxiliary structures in many disparate scientific fields, even unrelated to topology. For short, the notion is useful since it allows us to consider a relation of “being generated by” with no need to require the existence of an actual “closure” or “ hull”, which is problematic in certain contexts. 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. We notice that a categorical argument guarantees the existence of universal embeddings in many parallel situations.","PeriodicalId":41919,"journal":{"name":"Categories and General Algebraic Structures with Applications","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43144038","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove that the (b, c)-inverse and the inverse along an element in a semigroup are actually genuine inverse when considered as morphisms in the Schützenberger category of a semigroup. Applications to the Reverse Order Law are given. C Green’s relations and the Schützenberger category of a semigroup In this first section, we provide the reader with the necessary definitions and results regarding semigroups and categories. In particular, we recall the definition of the Schützenberger category of a semigroup and the interpretation of Green’s relations in this setting. Section 2 then presents the main result of the article (Theorem C.7), that (b, c)-inverses (and inverses along an element) are genuine inverses when considered as morphisms in the corresponding Schützenberger category. Finally, applications to the Reverse Order Law are given in Section 3.
{"title":"(b, c)-inverse, inverse along an element, and the Schützenberger category of a semigroup","authors":"X. Mary","doi":"10.52547/cgasa.15.1.255","DOIUrl":"https://doi.org/10.52547/cgasa.15.1.255","url":null,"abstract":"We prove that the (b, c)-inverse and the inverse along an element in a semigroup are actually genuine inverse when considered as morphisms in the Schützenberger category of a semigroup. Applications to the Reverse Order Law are given. C Green’s relations and the Schützenberger category of a semigroup In this first section, we provide the reader with the necessary definitions and results regarding semigroups and categories. In particular, we recall the definition of the Schützenberger category of a semigroup and the interpretation of Green’s relations in this setting. Section 2 then presents the main result of the article (Theorem C.7), that (b, c)-inverses (and inverses along an element) are genuine inverses when considered as morphisms in the corresponding Schützenberger category. Finally, applications to the Reverse Order Law are given in Section 3.","PeriodicalId":41919,"journal":{"name":"Categories and General Algebraic Structures with Applications","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70683546","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we study epimorphisms, dominions and related properties for some classes of structurally (n,m)-regular semigroups for any pair (n,m) of positive integers. In Section 2, after a brief introduction of these semigroups, we prove that the class of structurallly (n,m)-generalized inverse semigroups is closed under morphic images. We then prove the main result of this section that the class of structurally (n,m)-generalized inverse semigroups is saturated and, thus, in the category of all semigroups, epimorphisms in this class are precisely surjective morphisms. Finally, in the last section, we prove that the variety of structurally (o, n)-left regular bands is saturated in the variety of structurally (o, k)-left regular bands for all positive integers k and n with 1 6 k 6 n. C Introduction and preliminaries A morphism α : S → T in the category of all semigroups is called an epimorphism (epi for short) if for all morphisms β, γ with αβ = αγ implies * Corresponding author
{"title":"On epimorphisms and structurally regular semigroups","authors":"A. Shah, S. Bano, S. Ahanger, W. Ashraf","doi":"10.52547/cgasa.15.1.231","DOIUrl":"https://doi.org/10.52547/cgasa.15.1.231","url":null,"abstract":"In this paper we study epimorphisms, dominions and related properties for some classes of structurally (n,m)-regular semigroups for any pair (n,m) of positive integers. In Section 2, after a brief introduction of these semigroups, we prove that the class of structurallly (n,m)-generalized inverse semigroups is closed under morphic images. We then prove the main result of this section that the class of structurally (n,m)-generalized inverse semigroups is saturated and, thus, in the category of all semigroups, epimorphisms in this class are precisely surjective morphisms. Finally, in the last section, we prove that the variety of structurally (o, n)-left regular bands is saturated in the variety of structurally (o, k)-left regular bands for all positive integers k and n with 1 6 k 6 n. C Introduction and preliminaries A morphism α : S → T in the category of all semigroups is called an epimorphism (epi for short) if for all morphisms β, γ with αβ = αγ implies * Corresponding author","PeriodicalId":41919,"journal":{"name":"Categories and General Algebraic Structures with Applications","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70683641","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let C(L) be the ring of all continuous real functions on a frame L and S ⊆ R. An α ∈ C(L) is said to be an overlap of S, denoted by α J S, whenever u ∩ S ⊆ v ∩ S implies α(u) 6 α(v) for every open sets u and v in R. This concept was first introduced by A. Karimi-Feizabadi, A.A. Estaji, M. Robat-Sarpoushi in Pointfree version of image of real-valued continuous functions (2018). Although this concept is a suitable model for their purpose, it ultimately does not provide a clear definition of the range of continuous functions in the context of pointfree topology. In this paper, we will introduce a concept which is called pre-image, denoted by pim, as a pointfree version of the image of real-valued continuous functions on a topological space X. We investigate this concept and in addition to showing pim(α) = ⋂{S ⊆ R : α J S}, we will see that this concept is a good surrogate for the image of continuous real functions. For instance, we prove, under some achievable conditions, we have pim(α ∨ β) ⊆ pim(α) ∨ pim(β), pim(α ∧ β) ⊆ pim(α) ∧ pim(β), pim(αβ) ⊆ pim(α)pim(β) and pim(α+ β) ⊆ pim(α) + pim(β). * Corresponding author
{"title":"Pre-image of functions in $C(L)$","authors":"Ali Rezaei Aliabad, M. Mahmoudi","doi":"10.52547/cgasa.15.1.35","DOIUrl":"https://doi.org/10.52547/cgasa.15.1.35","url":null,"abstract":"Let C(L) be the ring of all continuous real functions on a frame L and S ⊆ R. An α ∈ C(L) is said to be an overlap of S, denoted by α J S, whenever u ∩ S ⊆ v ∩ S implies α(u) 6 α(v) for every open sets u and v in R. This concept was first introduced by A. Karimi-Feizabadi, A.A. Estaji, M. Robat-Sarpoushi in Pointfree version of image of real-valued continuous functions (2018). Although this concept is a suitable model for their purpose, it ultimately does not provide a clear definition of the range of continuous functions in the context of pointfree topology. In this paper, we will introduce a concept which is called pre-image, denoted by pim, as a pointfree version of the image of real-valued continuous functions on a topological space X. We investigate this concept and in addition to showing pim(α) = ⋂{S ⊆ R : α J S}, we will see that this concept is a good surrogate for the image of continuous real functions. For instance, we prove, under some achievable conditions, we have pim(α ∨ β) ⊆ pim(α) ∨ pim(β), pim(α ∧ β) ⊆ pim(α) ∧ pim(β), pim(αβ) ⊆ pim(α)pim(β) and pim(α+ β) ⊆ pim(α) + pim(β). * Corresponding author","PeriodicalId":41919,"journal":{"name":"Categories and General Algebraic Structures with Applications","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70683228","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The effect of the existence of a medial or related idempotent in any abundant semigroup is the subject of this paper. The aim is to naturally order any abundant semigroup S which contains an ample multiplicative medial idempotent u in a way that L∗ and R∗ are compatible with the natural order and u is a maximum idempotent. The structure of an abundant semigroup containing an ample normal medial idempotent studied in [6] will be revisited.
{"title":"Abundant semigroups with medial idempotents","authors":"A. El-Qallali","doi":"10.52547/cgasa.15.1.1","DOIUrl":"https://doi.org/10.52547/cgasa.15.1.1","url":null,"abstract":"The effect of the existence of a medial or related idempotent in any abundant semigroup is the subject of this paper. The aim is to naturally order any abundant semigroup S which contains an ample multiplicative medial idempotent u in a way that L∗ and R∗ are compatible with the natural order and u is a maximum idempotent. The structure of an abundant semigroup containing an ample normal medial idempotent studied in [6] will be revisited.","PeriodicalId":41919,"journal":{"name":"Categories and General Algebraic Structures with Applications","volume":"42 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70683497","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we study flatness properties (pullback flatness, limit flatness, finite limit flatness) of acts over semigroups. These are defined by requiring preservation of certain limits from the functor of tensor multiplication by a given act. We give a description of firm pullback flat acts using Conditions (P) and (E). We also study pure epimorphisms and their connections to finitely presented acts and pullback flat acts. We study these flatness properties in the category of all acts, as well as in the category of unitary acts and in the category of firm acts, which arise naturally in the Morita theory of semigroups.
{"title":"Flatness properties of acts over semigroups","authors":"V. Laan, Ülo Reimaa, L. Tart, Elery Teor","doi":"10.52547/cgasa.15.1.59","DOIUrl":"https://doi.org/10.52547/cgasa.15.1.59","url":null,"abstract":"In this paper we study flatness properties (pullback flatness, limit flatness, finite limit flatness) of acts over semigroups. These are defined by requiring preservation of certain limits from the functor of tensor multiplication by a given act. We give a description of firm pullback flat acts using Conditions (P) and (E). We also study pure epimorphisms and their connections to finitely presented acts and pullback flat acts. We study these flatness properties in the category of all acts, as well as in the category of unitary acts and in the category of firm acts, which arise naturally in the Morita theory of semigroups.","PeriodicalId":41919,"journal":{"name":"Categories and General Algebraic Structures with Applications","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70683305","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-12-14DOI: 10.52547/cgasa.2022.102443
Andrew Craig, B. Davey, M. Haviar
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's four values, for `true' ($t$), `false'($f$), `contradiction' ($top$) and `no information' ($bot$), 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, $mathbf J_n$, for $n in omega$, with $mathbf J_0$ being Belnap's seminal example. We gave a duality for the variety $mathcal V_n$ generated by $mathbf J_n$, with the objects of the dual category $mathcal X_n$ being multi-sorted topological structures. Here we study the dual category in depth. We give an axiomatisation of the category $mathcal X_n$ and show that it is isomorphic to a category $mathcal Y_n$ of single-sorted topological structures. The objects of $mathcal Y_n$ are Priestley spaces endowed with a continuous retraction in which the order has a natural ranking. We show how to construct the Priestley dual of the underlying bounded distributive lattice of an algebra in $mathcal V_n$ via its dual in $mathcal Y_n$; as an application we show that the size of the free algebra $mathbf F_{mathcal V_n}(1)$ is given by a polynomial in $n$ of degree $6$.
{"title":"Expanding Belnap 2: the dual category in depth","authors":"Andrew Craig, B. Davey, M. Haviar","doi":"10.52547/cgasa.2022.102443","DOIUrl":"https://doi.org/10.52547/cgasa.2022.102443","url":null,"abstract":"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's four values, for `true' ($t$), `false'($f$), `contradiction' ($top$) and `no information' ($bot$), 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. \u0000In our companion paper, we introduced a new family of prioritised default bilattices, $mathbf J_n$, for $n in omega$, with $mathbf J_0$ being Belnap's seminal example. We gave a duality for the variety $mathcal V_n$ generated by $mathbf J_n$, with the objects of the dual category $mathcal X_n$ being multi-sorted topological structures. \u0000Here we study the dual category in depth. We give an axiomatisation of the category $mathcal X_n$ and show that it is isomorphic to a category $mathcal Y_n$ of single-sorted topological structures. The objects of $mathcal Y_n$ are Priestley spaces endowed with a continuous retraction in which the order has a natural ranking. We show how to construct the Priestley dual of the underlying bounded distributive lattice of an algebra in $mathcal V_n$ via its dual in $mathcal Y_n$; as an application we show that the size of the free algebra $mathbf F_{mathcal V_n}(1)$ is given by a polynomial in $n$ of degree $6$.","PeriodicalId":41919,"journal":{"name":"Categories and General Algebraic Structures with Applications","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70684050","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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