Decompositions of universal algebras by idempotent algebras

Abstract

An algebra with units is an algebra in which every subalgebra contains a singleton subalgebra. A one-unit-algebra is an algebra in which every subalgebra contains exactly one singleton subalgebra. IfU,V are subclasses of a classK of algebras,UKV is the class of all\(\mathfrak{A}\)eK on which there is a congruence θ such that\(\mathfrak{A}\)/θeV and every θ-class that is a subalgebra of\(\mathfrak{A}\) belonging toK belongs also toU, e.g., ifK is the class of all semigroups,V is the class of all bands andU is the class of all groups,UKV is the class of all bands of groups. We studyUKV andUKU whereU is a class of one-unit-K-algebras andV is a class of idempotentK-algebras. IfK is a class of algebras of type τ closed under subalgebras and homomorphisms,U is the class of all one-unit-K-algebras andV is the class of all idempotentK-algebras, thenUKV is the class of allK-algebras that are τ-reducts of 〈τ, e〉-algebras\(\mathfrak{A}\) satisfying e(x) is a singleton subalgebra for everyx e A belonging to the τ-subalgebra of\(\mathfrak{A}\) generated byx and e(f(− x1, x2,..., xn))=e(fe(x1), e(x2),..., e(xn)) for every n-ary operationf of type τ. IfK is a variety of algebras with units and of finite type,U andV are finitely based (relative toK) subquasivarieties ofK, thenUKV is finitely based relative toK. IfK is the variety of all commutative groupoids with an additional unary operatione satisfying e(e(x))=e(x)=e(x)· e(x), e(x · y)=e(x)· e(y),U andV are the subvarities ofK defined by e(x)=e(y) andx=e(x) respectively, thenUKU is neither a variety nor finitely based. Some applications to semigroups and quasigroups are considered.