On involutive division on monoids

We consider an arbitrary monoid MM, on which an involutive division is introduced, and the set of all its finite subsets SetMM. Division is considered as a mapping d:SetMM{d:SetM \times M}, whose image d(U,m){d(U,m)} is the set of divisors of mm in UU. The properties of division and involutive divis...

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Autores principales:	Oleg K. Kroytor, Mikhail D. Malykh
Formato:	article
Lenguaje:	EN
Publicado:	Peoples’ Friendship University of Russia (RUDN University) 2021
Materias:	involutive monomial division gröbner basis Electronic computers. Computer science QA75.5-76.95
Acceso en línea:	https://doaj.org/article/c782bde967534217a915a46c36f1d9ab
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spelling	oai:doaj.org-article:c782bde967534217a915a46c36f1d9ab2021-11-12T15:18:25ZOn involutive division on monoids2658-46702658-714910.22363/2658-4670-2021-29-4-387-398https://doaj.org/article/c782bde967534217a915a46c36f1d9ab2021-12-01T00:00:00Zhttp://journals.rudn.ru/miph/article/viewFile/29430/20005https://doaj.org/toc/2658-4670https://doaj.org/toc/2658-7149We consider an arbitrary monoid MM, on which an involutive division is introduced, and the set of all its finite subsets SetMM. Division is considered as a mapping d:SetMM{d:SetM \times M}, whose image d(U,m){d(U,m)} is the set of divisors of mm in UU. The properties of division and involutive division are defined axiomatically. Involutive division was introduced in accordance with the definition of involutive monomial division, introduced by V.P. Gerdt and Yu.A. Blinkov. New notation is proposed that provides brief but explicit allowance for the dependence of division on the SetMM element. The theory of involutive completion (closures) of sets is presented for arbitrary monoids, necessary and sufficient conditions for completeness (closedness) - for monoids generated by a finite set XX. The analogy between this theory and the theory of completely continuous operators is emphasized. In the last section, we discuss the possibility of solving the problem of replenishing a given set by successively expanding the original domain and its connection with the axioms used in the definition of division. All results are illustrated with examples of Thomas monomial division.Oleg K. KroytorMikhail D. MalykhPeoples’ Friendship University of Russia (RUDN University)articleinvolutive monomial divisiongröbner basisElectronic computers. Computer scienceQA75.5-76.95ENDiscrete and Continuous Models and Applied Computational Science, Vol 29, Iss 4, Pp 387-398 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	involutive monomial division gröbner basis Electronic computers. Computer science QA75.5-76.95
spellingShingle	involutive monomial division gröbner basis Electronic computers. Computer science QA75.5-76.95 Oleg K. Kroytor Mikhail D. Malykh On involutive division on monoids
description	We consider an arbitrary monoid MM, on which an involutive division is introduced, and the set of all its finite subsets SetMM. Division is considered as a mapping d:SetMM{d:SetM \times M}, whose image d(U,m){d(U,m)} is the set of divisors of mm in UU. The properties of division and involutive division are defined axiomatically. Involutive division was introduced in accordance with the definition of involutive monomial division, introduced by V.P. Gerdt and Yu.A. Blinkov. New notation is proposed that provides brief but explicit allowance for the dependence of division on the SetMM element. The theory of involutive completion (closures) of sets is presented for arbitrary monoids, necessary and sufficient conditions for completeness (closedness) - for monoids generated by a finite set XX. The analogy between this theory and the theory of completely continuous operators is emphasized. In the last section, we discuss the possibility of solving the problem of replenishing a given set by successively expanding the original domain and its connection with the axioms used in the definition of division. All results are illustrated with examples of Thomas monomial division.
format	article
author	Oleg K. Kroytor Mikhail D. Malykh
author_facet	Oleg K. Kroytor Mikhail D. Malykh
author_sort	Oleg K. Kroytor
title	On involutive division on monoids
title_short	On involutive division on monoids
title_full	On involutive division on monoids
title_fullStr	On involutive division on monoids
title_full_unstemmed	On involutive division on monoids
title_sort	on involutive division on monoids
publisher	Peoples’ Friendship University of Russia (RUDN University)
publishDate	2021
url	https://doaj.org/article/c782bde967534217a915a46c36f1d9ab
work_keys_str_mv	AT olegkkroytor oninvolutivedivisiononmonoids AT mikhaildmalykh oninvolutivedivisiononmonoids
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On involutive division on monoids

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