Dominions in quasivarieties of universal algebras
Let \(\mathcal M\) be a class of algebras, and let \(H\) be a subalgebra of the algebra \(A\), then the dominion of \(H\) in \(A\) (in a class \(\mathcal M\)) is defined by: \[ \text{dom}^{\mathcal M}_A(H)=\{a\in A\mid \forall M\in {\mathcal M}, \forall f,g : A\to M (f| _h=g| _g \Rightarrow a^f=a^g)\}. \] The dominion is intimately connected with the free product of two copies of \(A\) with \(H\) amalgamated, for it is the preimage in \(A\) of the intersection of the two images of \(A\) in the free product. Another useful property of the dominion is that if a quasivariety \(\mathcal M\) is generated by the set \(M\), then \(\text{ dom}^{\mathcal M}_A(H)= \text{dom}^M_A(H)\), provided certain finiteness conditions hold. Also, if \(\mathcal M\) is a quasivariety, \(A\) an arbitary algebra, and \(H\) a finitely generated subalgebra of \(A\), then the set \(L(A,H,{\mathcal M})\) of dominions \(\{\text{dom}^{\mathcal N}_A(H)\}\), where \(\mathcal N\) is a subquasivariety of \(\mathcal M\), is a lattice. The paper ends with an investigation, using examples from group theory, as to whether any of the finiteness conditions in the last two results can be omitted. Three open problems are given.
scientific article; zbMATH DE number 2153438
scientific article