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,U _K V is the class of all$\mathfrak{A}$εK on which there is a congruence θ such that$\mathfrak{A}$/θεV 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,U _K V is the class of all bands of groups. We studyU _K V andU _K U 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, thenU _K V 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 ε A belonging to the τ-subalgebra of$\mathfrak{A}$ generated byx and e(f(− x₁, x₂,..., x_n))=e(fe(x₁), e(x₂),..., e(x_n)) 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, thenU _K V 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, thenU _K U is neither a variety nor finitely based. Some applications to semigroups and quasigroups are considered.