The lattice of subquasivarieties of a locally finite quasivariety
It was an old idea by G. Birkhoff and A. I. Malcev that quasivarieties could be investigated in terms of their lattices of subquasivarieties. This is just the approach taken by the authors. They restrict their attention to locally finite quasivarieties because this investigation aroses from their study of finitely generated quasivarieties. The authors analyse the structure of the lattice \(L_q(K)\) of subquasivarieties of a quasivariety \(K\). It is known that this lattice is both algebraic and dually algebraic, join semidistributive and fermentable. Quasivarieties are characterized which are completely join irreducible or completely meet irreducible elements of \(L_q(K)\), likewise quasivarieties that are completely join prime or completely meet prime. For a finite algebra \(T\) in \(K\), there is a finite set \(E(T)\) of quasi-equations such that, for any subquasivariety \(Q\) of \(K\), it is the case that \(T\) is not in \(Q\) if and only if \(Q\) satisfies some quasi-equation from \(E(T)\). For each quasi-equation \(e\) from \(E(T)\), there is a finite list \(U_1\),\dots,\(U_k\) of finite algebras in \(K\) such that an algebra \(S\) of \(K\) satisfies \(e\) if and only if \(S\) contains no \(U_i\) as subalgebra. Every subquasivariety of \(K\) is a meet of completely meet irreducible quasivarieties. For every completely meet irreducible quasivariety \(M\) in \(L_q(K)\), there is a finite algebra \(T\), not in \(M\), such that \(M\) is determined by some quasi-equation \(e\) in \(E(T)\). Hence, finitely based quasivarieties can be characterized in terms of omitting finitely many algebras. Those methods are illustrated by applying them to quasivarieties of 1-unary and 2-unary algebras, lattices, abelian groups, and pure relational structures whose relations are all unary. The book contains 7 chapters with results, one chapter with open problems, an appendix, a bibliography and an author index. The majority of this text is based on published results by V. Gorbunov, K. Adaricheva and J. B. Nation.
scientific article; zbMATH DE number 6863923