Freeoids: a semi-abstract view on endomorphism monoids of relatively free algebras

Abstract

A freeoid over a (normally, infinite) set of variables X is defined to be a pair (W, E), where W is a superset of X, and E is a submonoid of W W containing just one extension of every mapping X → W . For instance, if W is a relatively free algebra over a set of free generators X, then the pair F (W) := (W, End (W)) is a freeoid. In the paper, the kernel equivalence and the range of the transformation F are characterized. Freeoids form a category; it is shown that the transformation F gives rise to a functor from the category of relatively free algebras to the category of freeoids which yields a concrete equivalence of the first category to a full subcategory of the second one. Also, the concept of a model of a freeoid is introduced; the variety generated by a free algebra W is shown to be concretely equivalent to the category of models of F (W). The sets X, W , and the algebras W may generally be many-sorted.