Algebras with dimension (Q690115)

This is a study of the notions of dependence and dimension introduced by \textit{P. M. Cohn} [Universal algebra (1965; Zbl 0141.010); 2nd ed. (1981; Zbl 0461.08001)]. After some general definitions and remarks, the authors concentrate on so-called standard dependence relations defined as follows. For a given equational class \(\mathcal K\) and a \({\mathcal K}\)-algebra \(\mathcal A\), a subset \(B\) of \(\mathcal A\) is said to \({\mathcal K}\)-independent if it is either empty or it generates the subalgebra \([B]\) freely over \(\mathcal K\). The span \(\langle X\rangle\) of a subset \(X\) of \(\mathcal A\) consists then of the elements \(a(\in A)\) such that \(a\in X\) or \(Y\cup\{a\}\) is dependent for some independent subset \(Y\) of \(X\). A dependence is transitive if \(\langle\langle X\rangle\rangle= \langle X\rangle\) for all subsets \(X\). Two of the main theorems yield necessary and sufficient conditions for an equational class of unary algebras to have transitive standard dependences. The span of a subset of a \({\mathcal K}\)-algebra and the subalgebra generated by it turn out to be quite separate notions: neither one is necessarily even contained in the other. The authors also show that the transitivity of the standard dependence is not always preserved under direct products or homomorphisms, and they point out some mistakes in Cohn's book. Furthermore, some common constructions of automata theory are mentioned as the original motivation for this work.