Pub Date : 2022-11-30DOI: 10.1142/s0218196722500710
Song Juae
We prove that the rational function semifield of a tropical curve is finitely generated as a semifield over the tropical semifield [Formula: see text] by giving a specific finite generating set. Also, we show that for a finite harmonic morphism between tropical curves [Formula: see text], the rational function semifield of [Formula: see text] is finitely generated as a [Formula: see text]-algebra, where [Formula: see text] stands for the pull-back of the rational function semifield of [Formula: see text] by [Formula: see text].
{"title":"Rational function semifields of tropical curves are finitely generated over the tropical semifield","authors":"Song Juae","doi":"10.1142/s0218196722500710","DOIUrl":"https://doi.org/10.1142/s0218196722500710","url":null,"abstract":"We prove that the rational function semifield of a tropical curve is finitely generated as a semifield over the tropical semifield [Formula: see text] by giving a specific finite generating set. Also, we show that for a finite harmonic morphism between tropical curves [Formula: see text], the rational function semifield of [Formula: see text] is finitely generated as a [Formula: see text]-algebra, where [Formula: see text] stands for the pull-back of the rational function semifield of [Formula: see text] by [Formula: see text].","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":"20 1","pages":"1575-1594"},"PeriodicalIF":0.0,"publicationDate":"2022-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86987294","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 : 2022-11-28DOI: 10.1142/s0218196723500054
Craig Miller
The [Formula: see text]-height of a semigroup [Formula: see text] is the height of the poset of [Formula: see text]-classes of [Formula: see text] Given a semigroup [Formula: see text] with finite [Formula: see text]-height, we establish bounds on the [Formula: see text]-height of bi-ideals, one-sided ideals and two-sided ideals; in particular, these substructures inherit the property of having finite [Formula: see text]-height. We then investigate whether these bounds can be attained.
{"title":"The ℛ-height of semigroups and their bi-ideals","authors":"Craig Miller","doi":"10.1142/s0218196723500054","DOIUrl":"https://doi.org/10.1142/s0218196723500054","url":null,"abstract":"The [Formula: see text]-height of a semigroup [Formula: see text] is the height of the poset of [Formula: see text]-classes of [Formula: see text] Given a semigroup [Formula: see text] with finite [Formula: see text]-height, we establish bounds on the [Formula: see text]-height of bi-ideals, one-sided ideals and two-sided ideals; in particular, these substructures inherit the property of having finite [Formula: see text]-height. We then investigate whether these bounds can be attained.","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":"s3-23 1","pages":"47-66"},"PeriodicalIF":0.0,"publicationDate":"2022-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90814988","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 : 2022-11-24DOI: 10.1142/s0218196723500121
Milan Z. Grulović, Jelena Jovanovic, B. Seselja, A. Tepavčević
{"title":"Lattice characterization of some classes of groups by series of subgroups","authors":"Milan Z. Grulović, Jelena Jovanovic, B. Seselja, A. Tepavčević","doi":"10.1142/s0218196723500121","DOIUrl":"https://doi.org/10.1142/s0218196723500121","url":null,"abstract":"","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":"55 5","pages":"211-235"},"PeriodicalIF":0.0,"publicationDate":"2022-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72616739","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 : 2022-11-21DOI: 10.1142/s0218196723500078
Bertalan Bodor, Kende Kalina, Csaba A. Szabó
For an algebra [Formula: see text] the algebra [Formula: see text] is called a functional reduct if each [Formula: see text] is a term function of [Formula: see text]. We classify the functional reducts of the countable atomless Boolean algebra up to first-order interdefinability. That is, we consider two functional reducts the “same” if their group of automorphisms is the same. We show that there are 13 such reducts and describe their structures and group of automorphisms.
{"title":"Functional reducts of the countable atomless Boolean algebra","authors":"Bertalan Bodor, Kende Kalina, Csaba A. Szabó","doi":"10.1142/s0218196723500078","DOIUrl":"https://doi.org/10.1142/s0218196723500078","url":null,"abstract":"For an algebra [Formula: see text] the algebra [Formula: see text] is called a functional reduct if each [Formula: see text] is a term function of [Formula: see text]. We classify the functional reducts of the countable atomless Boolean algebra up to first-order interdefinability. That is, we consider two functional reducts the “same” if their group of automorphisms is the same. We show that there are 13 such reducts and describe their structures and group of automorphisms.","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":"15 1","pages":"87-103"},"PeriodicalIF":0.0,"publicationDate":"2022-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74337443","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 : 2022-11-21DOI: 10.1142/s0218196723500042
G. Gyenizse, M. Maróti, L. Zádori
We have recently published a result that [Formula: see text]-permutability is not join-prime in the lattice of interpretability types of varieties whenever [Formula: see text]. In the proof, we showed that if [Formula: see text], then the join of a properly chosen finitely generated non-[Formula: see text]-permutable variety and the variety [Formula: see text] defined by the majority identities is [Formula: see text]-permutable. In the present note, we prove that the join of any locally finite non-3-permutable variety with [Formula: see text] is non-3-permutable. We also prove that the join of any non-2-permutable variety with [Formula: see text] is non-2-permutable. Our non-3-permutable result gives that one has to use a nonlocally finite non-3-permutable variety [Formula: see text] if they want to prove that 3-permutability is not join-prime by arguing that [Formula: see text] is 3-permutable.
{"title":"On the use of majority for investigating primeness of 3-permutability","authors":"G. Gyenizse, M. Maróti, L. Zádori","doi":"10.1142/s0218196723500042","DOIUrl":"https://doi.org/10.1142/s0218196723500042","url":null,"abstract":"We have recently published a result that [Formula: see text]-permutability is not join-prime in the lattice of interpretability types of varieties whenever [Formula: see text]. In the proof, we showed that if [Formula: see text], then the join of a properly chosen finitely generated non-[Formula: see text]-permutable variety and the variety [Formula: see text] defined by the majority identities is [Formula: see text]-permutable. In the present note, we prove that the join of any locally finite non-3-permutable variety with [Formula: see text] is non-3-permutable. We also prove that the join of any non-2-permutable variety with [Formula: see text] is non-2-permutable. Our non-3-permutable result gives that one has to use a nonlocally finite non-3-permutable variety [Formula: see text] if they want to prove that 3-permutability is not join-prime by arguing that [Formula: see text] is 3-permutable.","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":"47 1","pages":"31-46"},"PeriodicalIF":0.0,"publicationDate":"2022-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88826168","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 : 2022-10-28DOI: 10.1142/s0218196723500066
J. Rosales, M. B. Branco, M. A. Traesel
{"title":"Numerical semigroups without consecutive small elements","authors":"J. Rosales, M. B. Branco, M. A. Traesel","doi":"10.1142/s0218196723500066","DOIUrl":"https://doi.org/10.1142/s0218196723500066","url":null,"abstract":"","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":"43 1","pages":"67-85"},"PeriodicalIF":0.0,"publicationDate":"2022-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77063178","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 : 2022-10-23DOI: 10.1142/s0218196723500182
Bakhyt Aitzhanova, U. Umirbaev
We prove that every derivation and every locally nilpotent derivation of the subalgebra $K[x^n, x^{n-1}y,ldots,xy^{n-1}, y^n]$, where $ngeq 2$, of the polynomial algebra $K[x,y]$ in two variables over a field $K$ of characteristic zero is induced by a derivation and a locally nilpotent derivation of $K[x,y]$, respectively. Moreover, we prove that every automorphism of $K[x^n, x^{n-1}y,ldots,xy^{n-1}, y^n]$ over an algebraically closed field $K$ of characteristic zero is induced by an automorphism of $K[x,y]$. We also show that the group of automorphisms of $K[x^n, x^{n-1}y,ldots,xy^{n-1}, y^n]$ admits an amalgamated free product structure.
{"title":"Automorphisms of affine Veronese surfaces","authors":"Bakhyt Aitzhanova, U. Umirbaev","doi":"10.1142/s0218196723500182","DOIUrl":"https://doi.org/10.1142/s0218196723500182","url":null,"abstract":"We prove that every derivation and every locally nilpotent derivation of the subalgebra $K[x^n, x^{n-1}y,ldots,xy^{n-1}, y^n]$, where $ngeq 2$, of the polynomial algebra $K[x,y]$ in two variables over a field $K$ of characteristic zero is induced by a derivation and a locally nilpotent derivation of $K[x,y]$, respectively. Moreover, we prove that every automorphism of $K[x^n, x^{n-1}y,ldots,xy^{n-1}, y^n]$ over an algebraically closed field $K$ of characteristic zero is induced by an automorphism of $K[x,y]$. We also show that the group of automorphisms of $K[x^n, x^{n-1}y,ldots,xy^{n-1}, y^n]$ admits an amalgamated free product structure.","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":"11 2 1","pages":"351-367"},"PeriodicalIF":0.0,"publicationDate":"2022-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77229931","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 : 2022-08-31DOI: 10.1142/s0218196722500643
K. Paykan
{"title":"Morita contexts and the projective socle property","authors":"K. Paykan","doi":"10.1142/s0218196722500643","DOIUrl":"https://doi.org/10.1142/s0218196722500643","url":null,"abstract":"","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":"124 1","pages":"1447-1457"},"PeriodicalIF":0.0,"publicationDate":"2022-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75361595","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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