L-UP AND MIRROR ALGEBRAS

In this paper we consider several families of abstract algebras including the well- known BCK-algebras and several larger classes including the class of d-algebras which is a generalization of BCK-algebras. For these algebras it is usually difficult and often impossible to obtain a complementation operation and the associated "de Morgan's laws". In this paper we construct a "mirror image" of a given algebra which when adjoined to the original algebra permit a natural complementation to take place. The class of BCK-algebras is not closed under this operation but the class of d-algebras is, thus explaining why it may be better to work with this class rather than the class of BCK-algebras. Other classes of interest in this setting are also discussed. abstract algebras: BCH-algebras. They have shown that the class of BCI-algebras is a proper subclass of the class of BCH-algebras. The present authors ((7)) introduced the notion of d-algebras which is another useful generalization of BCK-algebras, and then they investigated several relations between d-algebras and BCK-algebras as well as some other interesting relations between d-algebras and oriented digraphs. Recently, Y. B. Jun, E. H. Roh and H. S. Kim ((5)) introduced a new notion, called an BH-algebra, which is a general- ization of BCH/BCI/BCK-algebras, and defined the notions of ideals and boundedness in BH-algebras, and showed that there is a maximal ideal in bounded BH-algebras. Further- more, they constructed the quotient BH-algebras via translation ideals and obtained the fundamental theorem of homomorphisms for BH-algebras as a consequence. The present authors ((8)) gave an analytic method for constructing proper examples of a great variety of non-associative algebras of the BCK-type and generalizations of these. In this paper we consider several families of abstract algebras including the well-known BCK-algebras and several larger classes including the class of d-algebras which is a generalization of BCK- algebras. For these algebras it is usually difficult and often impossible to obtain a comple- mentation operation and the associated "de Morgan's laws". In this paper we construct a "mirror image" of a given algebra which when adjoined to the original algebra permit a natural complementation to take place. The class of BCK-algebras is not closed under this operation but the class of d-algebras is, thus explaining why it may be better to work with this class rather than the class of BCK-algebras. Other classes of interest in this setting are also discussed.