Abstract An intuitionistic fuzzy finite state automaton assigns a membership and nonmembership values in which there is a unique membership transition on an input symbol (IFAUM) is considered. It is proved and illustrated the existence of two different intuitionistic fuzzy monoids F (𝒜) and S𝒜 from an intuitionistic fuzzy transition function of the given IFAUM 𝒜. Also it is proved that F (𝒜) and S𝒜 are anti-isomorphic as monoids.
{"title":"Intuitionistic Fuzzy Monoids in an Intuitionistic Fuzzy Finite Automaton with Unique Membership Transition on an Input Symbol","authors":"K. Jency Priya, T. Rajaretnam","doi":"10.7151/dmgaa.1397","DOIUrl":"https://doi.org/10.7151/dmgaa.1397","url":null,"abstract":"Abstract An intuitionistic fuzzy finite state automaton assigns a membership and nonmembership values in which there is a unique membership transition on an input symbol (IFAUM) is considered. It is proved and illustrated the existence of two different intuitionistic fuzzy monoids F (𝒜) and S𝒜 from an intuitionistic fuzzy transition function of the given IFAUM 𝒜. Also it is proved that F (𝒜) and S𝒜 are anti-isomorphic as monoids.","PeriodicalId":36816,"journal":{"name":"Discussiones Mathematicae - General Algebra and Applications","volume":"42 1","pages":"383 - 394"},"PeriodicalIF":0.0,"publicationDate":"2022-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47212241","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}
Abstract In this paper, we consider Leavitt path algebras having coefficients in a k-semifield. Concentrating on the aspect of k-simplicity, we find a set of necessary and sufficient conditions for the k-simplicity of the Leavitt path algebra LS(Γ) of a directed graph Γ over a non-zeroid k-semifield S.
{"title":"k-Simplicity of Leavitt Path Algebras with Coefficients in a k-Semifield","authors":"Raibatak Sen Gupta, Mrinal K. Sen","doi":"10.7151/dmgaa.1388","DOIUrl":"https://doi.org/10.7151/dmgaa.1388","url":null,"abstract":"Abstract In this paper, we consider Leavitt path algebras having coefficients in a k-semifield. Concentrating on the aspect of k-simplicity, we find a set of necessary and sufficient conditions for the k-simplicity of the Leavitt path algebra LS(Γ) of a directed graph Γ over a non-zeroid k-semifield S.","PeriodicalId":36816,"journal":{"name":"Discussiones Mathematicae - General Algebra and Applications","volume":"42 1","pages":"241 - 253"},"PeriodicalIF":0.0,"publicationDate":"2022-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47916339","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}
Abstract H. Strietz proved in 1975 that the minimum size of a generating set of the partition lattice Part(n) on the n-element set (n ≥ 4) equals 4. This classical result forms the foundation for this study. Strietz's results have been echoed by L. Zádori (1983), who gave a new elegant proof confirming the outcome. Several studies have indeed emerged henceforth concerning four-element generating sets of partition lattices. More recently more studies have presented the approach for the lower bounds on the number λ(n) of the four-element generating sets of Part(n) and statistical approach to λ(n) for small values of n. Also, G. Czédli and the present author have recently proved that certain direct products of partition lattices are also 4-generated. In a recent paper, G. Czédli has shown that this result has connection with information theory. On this basis, here we give a lower bound on the number ν(n) of 4-element generating sets of the direct product Part(n) × Part(n + 1) for n ≥ 7 using the results from previous studies. For n = 1, . . . , 5, we use a computer-aided approach; it gives exact values for n = 1, 2, 3, 4 but we need a statistical method for n = 5.
{"title":"Lower Bound for the Number of 4-Element Generating Sets of Direct Products of Two Neighboring Partition Lattices","authors":"Lilian Oluoch, Amenah Al-Najafi","doi":"10.7151/dmgaa.1393","DOIUrl":"https://doi.org/10.7151/dmgaa.1393","url":null,"abstract":"Abstract H. Strietz proved in 1975 that the minimum size of a generating set of the partition lattice Part(n) on the n-element set (n ≥ 4) equals 4. This classical result forms the foundation for this study. Strietz's results have been echoed by L. Zádori (1983), who gave a new elegant proof confirming the outcome. Several studies have indeed emerged henceforth concerning four-element generating sets of partition lattices. More recently more studies have presented the approach for the lower bounds on the number λ(n) of the four-element generating sets of Part(n) and statistical approach to λ(n) for small values of n. Also, G. Czédli and the present author have recently proved that certain direct products of partition lattices are also 4-generated. In a recent paper, G. Czédli has shown that this result has connection with information theory. On this basis, here we give a lower bound on the number ν(n) of 4-element generating sets of the direct product Part(n) × Part(n + 1) for n ≥ 7 using the results from previous studies. For n = 1, . . . , 5, we use a computer-aided approach; it gives exact values for n = 1, 2, 3, 4 but we need a statistical method for n = 5.","PeriodicalId":36816,"journal":{"name":"Discussiones Mathematicae - General Algebra and Applications","volume":"42 1","pages":"327 - 338"},"PeriodicalIF":0.0,"publicationDate":"2022-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45471271","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}
Abstract Let (G, *) be a finite group and S = {u ∈ G|u ≠ u−1}, then the inverse graph is defined as a graph whose vertices coincide with G such that two distinct vertices u and v are adjacent if and only if either u * v ∈ S or v * u ∈ S. In this paper, we introduce a modified version of the inverse graph, called i*-graph associated with a group G. The i*-graph is a simple graph with vertex set consisting of elements of G and two vertices x, y ∈ Γ are adjacent if x and y are not inverses of each other. We study certain properties and characteristics of this graph. Some parameters of the i*-graph are also determined.
{"title":"On the Non-Inverse Graph of a Group","authors":"Javeria Amreen, S. Naduvath","doi":"10.7151/dmgaa.1392","DOIUrl":"https://doi.org/10.7151/dmgaa.1392","url":null,"abstract":"Abstract Let (G, *) be a finite group and S = {u ∈ G|u ≠ u−1}, then the inverse graph is defined as a graph whose vertices coincide with G such that two distinct vertices u and v are adjacent if and only if either u * v ∈ S or v * u ∈ S. In this paper, we introduce a modified version of the inverse graph, called i*-graph associated with a group G. The i*-graph is a simple graph with vertex set consisting of elements of G and two vertices x, y ∈ Γ are adjacent if x and y are not inverses of each other. We study certain properties and characteristics of this graph. Some parameters of the i*-graph are also determined.","PeriodicalId":36816,"journal":{"name":"Discussiones Mathematicae - General Algebra and Applications","volume":"42 1","pages":"315 - 325"},"PeriodicalIF":0.0,"publicationDate":"2022-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71123318","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}
Abstract An involutive pocrim is a resituated integral partially ordered commutative monoid with an involution operator, consider as an algebra. In this paper it is proved that the variety of a finitely generated by involutive pocrims of finite type has a finitely based equational theory. We also study the algebraic geometry over compete lattices and we investigate the properties of being equationally Noetherian and uω-compact over such lattices.
{"title":"Algebraic Geometry Over Complete Lattices and Involutive Pocrims","authors":"A. Molkhasi, K. Shum","doi":"10.7151/dmgaa.1394","DOIUrl":"https://doi.org/10.7151/dmgaa.1394","url":null,"abstract":"Abstract An involutive pocrim is a resituated integral partially ordered commutative monoid with an involution operator, consider as an algebra. In this paper it is proved that the variety of a finitely generated by involutive pocrims of finite type has a finitely based equational theory. We also study the algebraic geometry over compete lattices and we investigate the properties of being equationally Noetherian and uω-compact over such lattices.","PeriodicalId":36816,"journal":{"name":"Discussiones Mathematicae - General Algebra and Applications","volume":"44 2-3","pages":"339 - 347"},"PeriodicalIF":0.0,"publicationDate":"2022-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41289137","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}
Fabrice Tchoua Yinga, B. B. Koguep Njionou, Etienne R. Temgoua Alomo
Abstract In this paper, we introduce the concepts of n-fold obstinate ideals, n-fold normal ideals, n-fold fantastic ideals and n-fold involutive ideals in residuated lattices, state and prove some of their properties. Several characterizations of these notions are derived and the relations between those notions are investigated. Also, we construct the correspondence between the notions of n-fold ideal and n-fold filter in residuated lattices.
{"title":"n-Fold Fantastic and n-Fold Involutive Ideals in Bounded Commutative Residuated Lattices","authors":"Fabrice Tchoua Yinga, B. B. Koguep Njionou, Etienne R. Temgoua Alomo","doi":"10.7151/dmgaa.1396","DOIUrl":"https://doi.org/10.7151/dmgaa.1396","url":null,"abstract":"Abstract In this paper, we introduce the concepts of n-fold obstinate ideals, n-fold normal ideals, n-fold fantastic ideals and n-fold involutive ideals in residuated lattices, state and prove some of their properties. Several characterizations of these notions are derived and the relations between those notions are investigated. Also, we construct the correspondence between the notions of n-fold ideal and n-fold filter in residuated lattices.","PeriodicalId":36816,"journal":{"name":"Discussiones Mathematicae - General Algebra and Applications","volume":"42 1","pages":"363 - 381"},"PeriodicalIF":0.0,"publicationDate":"2022-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42749953","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}
Abstract In this paper we introduce Sheffer stroke BE-algebras (briefly, SBE-algebras) and investigate a relationship between SBE-algebras and BE-algebras. By presenting a SBE-filter, an upper set and a SBE-subalgebra on a SBE-algebra, it is shown that any SBE-filter of a SBE-algebra is a SBE-subalgebra but the converse of this statement is not true. Besides we construct quotient SBE-algebras via a congruence relation defined by a special SBE-filter. We discuss SBE-homomorphisms and their properties between SBE-algebras. Finally, a relation between Sheffer stroke Hilbert algebras and SBE-algebras is established.
{"title":"On Sheffer Stroke Be-Algebras","authors":"T. Katican, T. Oner, A. Saeid","doi":"10.7151/dmgaa.1391","DOIUrl":"https://doi.org/10.7151/dmgaa.1391","url":null,"abstract":"Abstract In this paper we introduce Sheffer stroke BE-algebras (briefly, SBE-algebras) and investigate a relationship between SBE-algebras and BE-algebras. By presenting a SBE-filter, an upper set and a SBE-subalgebra on a SBE-algebra, it is shown that any SBE-filter of a SBE-algebra is a SBE-subalgebra but the converse of this statement is not true. Besides we construct quotient SBE-algebras via a congruence relation defined by a special SBE-filter. We discuss SBE-homomorphisms and their properties between SBE-algebras. Finally, a relation between Sheffer stroke Hilbert algebras and SBE-algebras is established.","PeriodicalId":36816,"journal":{"name":"Discussiones Mathematicae - General Algebra and Applications","volume":"42 1","pages":"293 - 314"},"PeriodicalIF":0.0,"publicationDate":"2022-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41384551","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}
Abstract As a generalization of the concept of a weakly prime ideal, we introduce the concepts of a fuzzy weak prime ideal, a fuzzy weakly 2-absorbing ideal of a lattice. Some results of fuzzy weakly 2-absorbing ideals and fuzzy weakly primary ideals are proved. We also introduce and study fuzzy weakly 2-absorbing ideals in a product of lattices.
{"title":"Fuzzy Weakly 2-Absorbing Ideals of a Lattice","authors":"S. Nimbhorkar, Y. Patil","doi":"10.7151/dmgaa.1389","DOIUrl":"https://doi.org/10.7151/dmgaa.1389","url":null,"abstract":"Abstract As a generalization of the concept of a weakly prime ideal, we introduce the concepts of a fuzzy weak prime ideal, a fuzzy weakly 2-absorbing ideal of a lattice. Some results of fuzzy weakly 2-absorbing ideals and fuzzy weakly primary ideals are proved. We also introduce and study fuzzy weakly 2-absorbing ideals in a product of lattices.","PeriodicalId":36816,"journal":{"name":"Discussiones Mathematicae - General Algebra and Applications","volume":"42 1","pages":"255 - 277"},"PeriodicalIF":0.0,"publicationDate":"2022-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48874904","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}
Abstract In this paper, we introduce a new family of circulants GA(t, k), called Generalized Andrásfai graphs, where t, k ≥ 2 are integers. We study various parameters like diameter, girth, domination number etc. of GA(t, k). Moreover, we find the full automorphism group of GA(t, k) and compute its determining number.
{"title":"Generalized Andrásfai Graphs","authors":"Sucharita Biswas, Angsuman Das, M. Saha","doi":"10.7151/dmgaa.1401","DOIUrl":"https://doi.org/10.7151/dmgaa.1401","url":null,"abstract":"Abstract In this paper, we introduce a new family of circulants GA(t, k), called Generalized Andrásfai graphs, where t, k ≥ 2 are integers. We study various parameters like diameter, girth, domination number etc. of GA(t, k). Moreover, we find the full automorphism group of GA(t, k) and compute its determining number.","PeriodicalId":36816,"journal":{"name":"Discussiones Mathematicae - General Algebra and Applications","volume":"42 1","pages":"449 - 462"},"PeriodicalIF":0.0,"publicationDate":"2022-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46085482","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}
Abstract The bihyperbolic numbers are extension of hyperbolic numbers to four dimensions. In this paper we introduce and study the Fibonacci and Lucas bihypernomials, i.e., polynomials, which are a generalization of the bihyperbolic Fibonacci numbers and the bihyperbolic Lucas numbers, respectively.
{"title":"A Study on Fibonacci and Lucas Bihypernomials","authors":"A. Szynal-Liana, I. Włoch","doi":"10.7151/dmgaa.1399","DOIUrl":"https://doi.org/10.7151/dmgaa.1399","url":null,"abstract":"Abstract The bihyperbolic numbers are extension of hyperbolic numbers to four dimensions. In this paper we introduce and study the Fibonacci and Lucas bihypernomials, i.e., polynomials, which are a generalization of the bihyperbolic Fibonacci numbers and the bihyperbolic Lucas numbers, respectively.","PeriodicalId":36816,"journal":{"name":"Discussiones Mathematicae - General Algebra and Applications","volume":"42 1","pages":"409 - 423"},"PeriodicalIF":0.0,"publicationDate":"2022-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47469050","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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