In this paper, after recalling the category {bf PosAct}-$S$ of all poset acts over a pomonoid $S$; an $S$-act in the category {bf Pos} of all posets, with action preserving monotone maps between them, some categorical properties of the category {bf PosAct}-$S$ are considered. In particular, we describe limits and colimits such as products, coproducts, equalizers, coequalizers and etc. in this category. Also, several kinds of epimorphisms and monomorphisms are characterized in {bf PosAct}-$S$. Finally, we study injectivity and projectivity in {bf PosAct}-$S$ with respect to (regular) monomorphisms and (regular) epimorphisms, respectively, and see that although there is no non-trivial injective poset act with respect to monomorphisms, {bf PosAct}-$S$ has enough regular injectives with respect to regular monomorphisms. Also, it is proved that regular injective poset acts are exactly retracts of cofree poset acts over complete posets.
{"title":"Injective and projective poset acts","authors":"L. Shahbaz","doi":"10.56415/qrs.v31.11","DOIUrl":"https://doi.org/10.56415/qrs.v31.11","url":null,"abstract":"In this paper, after recalling the category {bf PosAct}-$S$ of all poset acts over a pomonoid $S$; an $S$-act in the category {bf Pos} of all posets, with action preserving monotone maps between them, some categorical properties of the category {bf PosAct}-$S$ are considered. In particular, we describe limits and colimits such as products, coproducts, equalizers, coequalizers and etc. in this category. Also, several kinds of epimorphisms and monomorphisms are characterized in {bf PosAct}-$S$. Finally, we study injectivity and projectivity in {bf PosAct}-$S$ with respect to (regular) monomorphisms and (regular) epimorphisms, respectively, and see that although there is no non-trivial injective poset act with respect to monomorphisms, {bf PosAct}-$S$ has enough regular injectives with respect to regular monomorphisms. Also, it is proved that regular injective poset acts are exactly retracts of cofree poset acts over complete posets.","PeriodicalId":38681,"journal":{"name":"Quasigroups and Related Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45722529","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}
This work addresses some relevant characteristics and properties of anticenter-symmetric Jacobi-Jordan algebras such as bimodules, matched pairs. Besides, the Jacobi-Jordan admissible algebra is defined; a special emphasis is given to a double construction of quadratic anticenter-symmetric algebras. We then follow this theory with the main properties and related algebraic structures of an anticenter-symmetric JJ algebra, namely the anti-Zinbiel algebras. Finally, we discuss the double construction of some classes of the two dimensional anticenter-symmetric JJ algebras.
{"title":"A double construction of quadratic anticenter-symmetric Jacobi-Jordan algebras","authors":"Essossolim Cyrille Haliya, Gbevewou Damien Houndedji","doi":"10.56415/qrs.v31.03","DOIUrl":"https://doi.org/10.56415/qrs.v31.03","url":null,"abstract":"This work addresses some relevant characteristics and properties of anticenter-symmetric Jacobi-Jordan algebras such as bimodules, matched pairs. Besides, the Jacobi-Jordan admissible algebra is defined; a special emphasis is given to a double construction of quadratic anticenter-symmetric algebras. We then follow this theory with the main properties and related algebraic structures of an anticenter-symmetric JJ algebra, namely the anti-Zinbiel algebras. Finally, we discuss the double construction of some classes of the two dimensional anticenter-symmetric JJ algebras.","PeriodicalId":38681,"journal":{"name":"Quasigroups and Related Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49617021","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 introduce the notions of interior filters, quasi-interior filters and weak-interior filters in a quasi-ordered semigroup. Additionally, we study the properties of these types of filters of quasi-ordered semigroups and their interrelationships.
{"title":"Some types of interior filters in quasi-ordered semigroups","authors":"D. Romano","doi":"10.56415/qrs.v31.10","DOIUrl":"https://doi.org/10.56415/qrs.v31.10","url":null,"abstract":"In this paper, we introduce the notions of interior filters, quasi-interior filters and weak-interior filters in a quasi-ordered semigroup. Additionally, we study the properties of these types of filters of quasi-ordered semigroups and their interrelationships.","PeriodicalId":38681,"journal":{"name":"Quasigroups and Related Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42431670","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 An be the set of n X n zero-one matrices satisfying the matrix equation A2 = Jn; where Jn is n X n matrices of all ones. In this article, it is proved that the number of non-isomorphic left loops of order k gives the lower bound to the size of An for n = k2. Mainly we have established that any matrix in An corresponding to loop has rank 2k - 2, where n = k2, for some positive integer k.
{"title":"On central digraphs constructed from left loops and loops","authors":"Rajaram Rawat","doi":"10.56415/qrs.v31.09","DOIUrl":"https://doi.org/10.56415/qrs.v31.09","url":null,"abstract":"Let An be the set of n X n zero-one matrices satisfying the matrix equation A2 = Jn; where Jn is n X n matrices of all ones. In this article, it is proved that the number of non-isomorphic left loops of order k gives the lower bound to the size of An for n = k2. Mainly we have established that any matrix in An corresponding to loop has rank 2k - 2, where n = k2, for some positive integer k.","PeriodicalId":38681,"journal":{"name":"Quasigroups and Related Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48270515","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 extend the classical theorem of R.Ellis to completely inverse $ AG^{astast}$- groupoids and we describe topologies on $ AG^{astast} $-groupoid induced by family of pseudometrics.
{"title":"On topological completely inverse AG**-groupoids","authors":"Hamza Boujouf","doi":"10.56415/qrs.v31.02","DOIUrl":"https://doi.org/10.56415/qrs.v31.02","url":null,"abstract":"We extend the classical theorem of R.Ellis to completely inverse $ AG^{astast}$- groupoids and we describe topologies on $ AG^{astast} $-groupoid induced by family of pseudometrics.","PeriodicalId":38681,"journal":{"name":"Quasigroups and Related Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45795859","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}
Richard Ilemobade, Olufemi George, Jaiyeola Temitope Gbolahan
This paper introduced a condition called $mathcal{R}$-condition under which $(r,s,t)$-inverse quasigroups are universal. Middle isotopic $(r,s,t)$-inverse loops, satisfying the $mathcal{R}$-condition and possessing a trivial set of $r$-weak inverse permutations were shown to be isomorphic; isotopy-isomorphy for $(r,s,t)$-inverse loops. Isotopy-isomorphy for $(r,s,t)$-inverse loops was generally characterized. With the $mathcal{R}$-condition, it was shown that for positive integers $r$, $s$ and $t$, if there is a $(r,s,t)$-inverse quasigroup of order $3k$ with an inverse-cycle of length $gcd(k,r+s+t)>1$, then there exists an $(r,s,t)$-inverse quasigroup of order $3k$ with an inverse-cycle of length $gcdbig(k(r+s+t), (r+s+t)^2big)$. The procedure of application of such $(r,s,t)$-inverse quasigroups to cryptography was described and explained, while the feasibility of such $(r,s,t)$-inverse quasigroups was illustrated with sample values of $k,r,s$ and $t$.
{"title":"On the universality and isotopy-isomorphy of (r,s,t)-inverse quasigroups and loops with applications to cryptography","authors":"Richard Ilemobade, Olufemi George, Jaiyeola Temitope Gbolahan","doi":"10.56415/qrs.v31.04","DOIUrl":"https://doi.org/10.56415/qrs.v31.04","url":null,"abstract":"This paper introduced a condition called $mathcal{R}$-condition under which $(r,s,t)$-inverse quasigroups are universal. Middle isotopic $(r,s,t)$-inverse loops, satisfying the $mathcal{R}$-condition and possessing a trivial set of $r$-weak inverse permutations were shown to be isomorphic; isotopy-isomorphy for $(r,s,t)$-inverse loops. Isotopy-isomorphy for $(r,s,t)$-inverse loops was generally characterized. With the $mathcal{R}$-condition, it was shown that for positive integers $r$, $s$ and $t$, if there is a $(r,s,t)$-inverse quasigroup of order $3k$ with an inverse-cycle of length $gcd(k,r+s+t)>1$, then there exists an $(r,s,t)$-inverse quasigroup of order $3k$ with an inverse-cycle of length $gcdbig(k(r+s+t), (r+s+t)^2big)$. The procedure of application of such $(r,s,t)$-inverse quasigroups to cryptography was described and explained, while the feasibility of such $(r,s,t)$-inverse quasigroups was illustrated with sample values of $k,r,s$ and $t$.","PeriodicalId":38681,"journal":{"name":"Quasigroups and Related Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47328258","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}
Finite semisymmetric quasigroups are in bijection with certain mappings between abstract polyhedra and directed graphs, termed alignments. We demonstrate the polyhedra of any given alignment can always be realized as compact, orientable surfaces. For any n to N, the class of quasigroups having associated surfaces with sum genus ≤ n is closed under subobjects and homomorphic images. Further, we demonstrate semisymmetric quasigroup homomorphisms may be translated into branched covers between their respective surfaces.
{"title":"Branched covers induced by semisymmetric quasigroup homomorphisms","authors":"Kyle M. Lewis","doi":"10.56415/qrs.v31.06","DOIUrl":"https://doi.org/10.56415/qrs.v31.06","url":null,"abstract":"Finite semisymmetric quasigroups are in bijection with certain mappings between abstract polyhedra and directed graphs, termed alignments. We demonstrate the polyhedra of any given alignment can always be realized as compact, orientable surfaces. For any n to N, the class of quasigroups having associated surfaces with sum genus ≤ n is closed under subobjects and homomorphic images. Further, we demonstrate semisymmetric quasigroup homomorphisms may be translated into branched covers between their respective surfaces.","PeriodicalId":38681,"journal":{"name":"Quasigroups and Related Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41264091","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}
Wichayaporn Jantanan, Chinnawat Jumnongphan, Natthawut Jaichot, R. Chinram
In this paper, we characterize when the radical $sqrt{I}$ of every interior ideal $I$ of a semigroup $S$ is a subsemigroup of $S$. Also, the radical of every interior ideal (or right ideal or left ideal or quasi-ideal or ideal or bi-ideal or subsemigroup) of $S$ is an interior ideal (or a right ideal or a left ideal or a quasi-ideal or an ideal or a bi-ideal) of $S$.
{"title":"Semigroups in which the radical of every interior ideal is a subsemigroup","authors":"Wichayaporn Jantanan, Chinnawat Jumnongphan, Natthawut Jaichot, R. Chinram","doi":"10.56415/qrs.v31.05","DOIUrl":"https://doi.org/10.56415/qrs.v31.05","url":null,"abstract":"In this paper, we characterize when the radical $sqrt{I}$ of every interior ideal $I$ of a semigroup $S$ is a subsemigroup of $S$. Also, the radical of every interior ideal (or right ideal or left ideal or quasi-ideal or ideal or bi-ideal or subsemigroup) of $S$ is an interior ideal (or a right ideal or a left ideal or a quasi-ideal or an ideal or a bi-ideal) of $S$.","PeriodicalId":38681,"journal":{"name":"Quasigroups and Related Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42694591","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 introduce the notion of weakly quasi invo-clean rings where every element $ r $ can be written as $ r=v+e $ or $ r=v-e $, where $vin Qinv(R)$ and $ ein Id(R) $. We study various properties of weakly quasi invo-clean elements and weakly quasi invo-clean rings. We prove that the ring $ R=prod_{iin I} R_i $, where all rings $ R_i $ are weakly quasi invo-clean, is weakly quasi invo-clean ring if and only if all factors but one are quasi invo-clean.
{"title":"Weakly quasi invo-clean rings","authors":"F. Rashedi","doi":"10.56415/qrs.v31.08","DOIUrl":"https://doi.org/10.56415/qrs.v31.08","url":null,"abstract":"We introduce the notion of weakly quasi invo-clean rings where every element $ r $ can be written as $ r=v+e $ or $ r=v-e $, where $vin Qinv(R)$ and $ ein Id(R) $. We study various properties of weakly quasi invo-clean elements and weakly quasi invo-clean rings. We prove that the ring $ R=prod_{iin I} R_i $, where all rings $ R_i $ are weakly quasi invo-clean, is weakly quasi invo-clean ring if and only if all factors but one are quasi invo-clean.","PeriodicalId":38681,"journal":{"name":"Quasigroups and Related Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42866393","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 analysis and recognition of fractal image patterns derived from Cayley tables has turned out to play a relevant role for distributing distinct types of algebraic and combinatorial structures into isomorphism classes. In this regard, Dimitrova and Markovski described in 2007 a graphical representation of quasigroups by means of fractal image patterns. It is based on the construction of pseudo-random sequences arising from a given quasigroup. In particular, isomorphic quasigroups give rise to the same fractal image pattern, up to permutation of underlying colors. This possible difference may be avoided by homogenizing the standard sets related to these patterns. Based on the differential box-counting method, the mean fractal dimension of homogenized standard sets constitutes a quasigroup isomorphism invariant which is analyzed in this paper in order to distribute quasigroups of the same order into isomorphism classes.
{"title":"A new quasigroup isomorphism invariant arising from fractal image patterns","authors":"Raúl M. Falcón","doi":"10.56415/qrs.v30.06","DOIUrl":"https://doi.org/10.56415/qrs.v30.06","url":null,"abstract":"The analysis and recognition of fractal image patterns derived from Cayley tables has turned out to play a relevant role for distributing distinct types of algebraic and combinatorial structures into isomorphism classes. In this regard, Dimitrova and Markovski described in 2007 a graphical representation of quasigroups by means of fractal image patterns. It is based on the construction of pseudo-random sequences arising from a given quasigroup. In particular, isomorphic quasigroups give rise to the same fractal image pattern, up to permutation of underlying colors. This possible difference may be avoided by homogenizing the standard sets related to these patterns. Based on the differential box-counting method, the mean fractal dimension of homogenized standard sets constitutes a quasigroup isomorphism invariant which is analyzed in this paper in order to distribute quasigroups of the same order into isomorphism classes.","PeriodicalId":38681,"journal":{"name":"Quasigroups and Related Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42832600","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：