- Strongly Solid Varieties and Free Generalized Clones
- Denecke, Klaus ;
- Kyungpook mathematical journal, volume 45, issue 1, 2005, Pages 33~33
Abstract
Clones are sets of operations which are closed under composition and contain all projections. 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 ([7]). 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 so-called generalized clones, to strong hyperidentities and to strongly solid varieties. On the basis of a generalized superposition operation for terms we generalize the concept of a unitary Menger algebra of finite rank ([6]) to unitary Menger algebras with infinitely many nullary operations and prove that strong hyperidentities correspond to identities in free unitary Menger algebras with infinitely many nullary operations.