Open AccessOn involutive division on monoids
Author(s) -
Oleg K. Kroytor,
М. Д. Малых
Publication year - 2021
Publication title -
discrete and continuous models and applied computational science
Language(s) - English
Resource type - Journals
eISSN - 2658-7149
pISSN - 2658-4670
DOI - 10.22363/2658-4670-2021-29-4-387-398
Subject(s) - division (mathematics) , mathematics , monomial , monoid , axiom , pure mathematics , algebra over a field , element (criminal law) , completeness (order theory) , notation , discrete mathematics , arithmetic , mathematical analysis , geometry , political science , law
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.