Constructions of non-commutative algebras. (English) Zbl 0597.08011

Universal algebra, Colloq., Szeged/Hung. 1983, Colloq. Math. Soc. János Bolyai 43, 177-187 (1986).
[For the entire collection see Zbl 0588.00012.]
Let (A,F) be a universal algebra without nullary operations, and \((S_ a\), \(a\in A)\) a collection of pairwise disjoint sets. Suppose that any \(f\in F_ n\) defines a map \(\bar f:\) \(S_ a\to S_{a'}\) where \(a'=f(a,b)\) for some \(b\in A^{n-1}\). If \(G=\cup S_ a\), \(f\in F_ n\), and \(b_ i\in S_{a_ i}\), \(1\leq i\leq n\), define \(f(b_ 1,...,b_ n)=\bar f(b_ 1)\in S_ a\), where \(a=f(a_ 1,...,a_ n)\). This construction is opposite to the construction of an homomorphic image and is close to a construction of aggasiz-sums by J. Plonka. If K is a class of F-algebras denote by G(K) the class of F-algebras G which are obtained by this construction from algebras A in K. If K is closed under subalgebras (direct products) then so is G(K). In general \(G(G(K))=G(K)\). If K is a variety, then H(G(K)) is a variety. There exists a variety V for which G(V) is not a variety.
Reviewer: V.A.Artamonov


08B25 Products, amalgamated products, and other kinds of limits and colimits
08A02 Relational systems, laws of composition


Zbl 0588.00012