Amalgamation classes of some distributive varieties
Let \({\mathcal H}\) be a class of algebras. Then a diagram in \({\mathcal H}\) is a quintuple \(<A,B_ 0,B_ 1,\alpha_ 0,\alpha_ 1>\) where \(A,B_ 0,B_ 1\in {\mathcal H}\) and \(\alpha_ 0\), \(\alpha_ 1\) are embeddings of A into \(B_ 0\), \(B_ 1\) respectively. The diagram is amalgamated in \({\mathcal H}\) by \(<C,\beta_ 0,\beta_ 1>\) if \(C\in {\mathcal H}\) and \(\beta_ 0\), \(\beta_ 1\) are embeddings of \(B_ 0\), \(B_ 1\) respectively into C such that \(\beta_ 0\circ \alpha_ 0=\beta_ 1\circ \alpha_ 1\). A is an amalgamation base of \({\mathcal H}\) if every \(<A,B_ 0,B_ 1,\alpha_ 0,\alpha_ 1>\) in \({\mathcal H}\) can be amalgamated in \({\mathcal H}\). The amalgamation class of \({\mathcal H}\), AMAL(\({\mathcal H})\) is the collection of all amalgamation bases of \({\mathcal H}\) and \({\mathcal H}\) has the amalgamation property of \({\mathcal H}=AMAL({\mathcal H}).\) As is well-known, very few varieties of algebras have the amalgamation property; this paper investigates the properties of amalgamation bases, particularly in varieties of lattices. The specific results obtained require too many technical definitions to be stated explicitly here.
scientific article; zbMATH DE number 3912451
scientific article