Abstract
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.
Highlights
The creation of the technique of involutive bases as an alternative to the classical Gröbner bases and its application to the study of ideals in polynomial and differential rings is undoubtedly one of the most important contributions made by V
The concept of involutive division came to algebras from the compatibility studies of systems of partial differential equations, dating back to the works of Riquier [1], Janet [2], Thomas [3]
In [13] it was noted that “involutive division for a monomial can be specified by defining sets of multiplicative and non-multiplicative variables”
Summary
The creation of the technique of involutive bases as an alternative to the classical Gröbner bases and its application to the study of ideals in polynomial and differential rings is undoubtedly one of the most important contributions made by V. The concept of involutive division came to algebras from the compatibility studies of systems of partial differential equations, dating back to the works of Riquier [1], Janet [2], Thomas [3]. The concept of involutive division was introduced by Gerdt and Blinkov in [8]–[11]. A number of terms (continuity of division, closure of sets) refer to this so far unknown topology. In our opinion, this topology is the Zariski topology, and incidental analogies taken from the R topology greatly hinder its development
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