Embedding modes into semimodules, Part I

In the first part of this paper, we considered the prob- lem of constructing a (commutative unital) semiring defining the variety of semimodules whose idempotent subreducts lie in a given variety of modes. We provided a general construction of such semirings, along with basic examples and some general proper- ties. In the second part of the paper we discussed some selected varieties of modes, in particular, varieties of affine spaces, vari- eties of barycentric algebras and varieties of semilattice modes, and described the semirings determining their semi-linearizations, the varieties of semimodules having these algebras as idempotent subreducts. The third part is devoted to varieties of differential groupoids and more general differential modes, and provides the semirings of the semi-linearizations of these varieties. This paper is a direct continuation of the first and second parts appearing with the same title (4) and (5). In the first part, we considered the problem of constructing a (commutative unital) semiring defining the variety of semimodules whose idempotent subreducts lie in a given variety of modes, and such that each semimodule-embeddable member of this mode variety embeds into a semimodule over the constructed semiring. We described a general construction of such semirings, with basic examples and some general properties. In the second part, we investigated selected varieties of modes, and described the semirings determining varieties of semimodules having algebras of these classes as subreducts, and discussed properties of the corresponding semi-affine spaces. In particular, we investigated varieties of affine spaces, varieties of barycentric algebras and varieties of semilattice modes. The third part is devoted to varieties of differential groupoids and more general