• Kyungpook Mathematical Journal
• /
• v.51 no.2
• /
• pp.139-143
• /
• 2011

In this paper we consider mappings ${\sigma}$ which map the binary operation symbol f to the term ${\sigma}$(f) which do not necessarily preserve the arities. We call these mappings generalized hypersubstitutions. Any generalized hypersubstitution ${\sigma}$ can be extended to a mapping $\hat{\sigma}$ on the set of all terms of type ${\tau}$ = (2). We de ne a binary operation on the set $Hyp_G$(2) of all generalized hypersubstitutions of type ${\tau}$ = (2) by using this extension The set $Hyp_G$(2) together with the identity generalized hypersubstitution ${\sigma}_{id}$ which maps f to the term f($x_1,x_2$) forms a monoid. We determine all regular elements of this monoid.

• Kyungpook Mathematical Journal
• /
• v.45 no.1
• /
• pp.33-43
• /
• 2005

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.

• Kyungpook Mathematical Journal
• /
• v.54 no.3
• /
• pp.501-509
• /
• 2014

In 2000, K. Denecke and K. Mahdavi showed that there are many idempotent elements in $Hyp_{N_{\varphi}}(V)$ the set of normal form hypersubstitutions of type ${\tau}=(2)$ which are not idempotent elements in Hyp(2) the set of all hypersubstitutions of type ${\tau}=(2)$. They considered in which varieties, idempotent elements of Hyp(2) are idempotent elements of $Hyp_{N_{\varphi}}(V)$. In this paper, we study the similar problems on the set of all generalized hypersubstitutions of type ${\tau}=(2)$ and the set of all normal form generalize hypersubstitutions of type ${\tau}=(2)$ and determine the order of normal form generalize hypersubstitutions of type ${\tau}=(2)$.