Menger Algebras and Clones of Terms

Abstract

Clones are sets of operations which are closed under superposition and contain all projections. The superposition operation maps to each (n + 1)−tuple of n-ary operations a new n-ary operation and satisfies the so-called superassociative law. The corresponding algebraic structures are Menger algebras of rank n, unitary Menger algebras of rank n and Menger algebras with infinitely many nullary operations. Identities of clones of term operations of a given algebra correspond to hyperidentities of this algebra, i.e. to identities which are satisfied after any replacements of fundamental operations by derived operations ([10]). If any identity of an algebra is satisfied as a hyperidentity, the algebra is called solid ([3]). Solid algebras correspond to free clones. These connections will be extended to strongly full clones, togeneralized clones, to strong hyperidentities and to strongly solid varieties. We prove that nhyperidentities,SF-hyperidentitiesand strong hyperidentities correspond to identities in free unitary Menger algebras of finite rank, in Menger algebras of finite rank or to free unitary Menger algebras with infinitely many nullary operations, respectively.