On BCK algebras: Part II: New algebras. The ordinal sum (product) of two bounded BCK algebras

Abstract

Since all the algebras connected to logic have, more or less explicitly, an associated order relation, it follows, by duality principle, that they have two presentations, dual to each other. We classify these dual presentations in "left" and "right" ones and we consider that, when dealing with several algebras in the same research, it is useful to present them unitarily, either as "left" algebras or as "right" algebras. In some circumstances, this choice is essential, for instance if we want to build the ordinal sum (product) between a BL algebra and an MV algebra. We have chosen the "left" presentation and several algebras of logic have been redefined as particular cases of BCK algebras. We introduce several new properties of algebras of logic, besides those usually existing in the literature, which generate a more refined classification, depending on the properties satisfied. In this work (Parts I---V) we make an exhaustive study of these algebras--with two bounds and with one bound--and we present classes of finite examples, in bounded case. In Part II, we continue to present new properties, and consequently new algebras; among them, bounded ? ? algebra is a common generalization of MTL algebra and divisible bounded residuated lattice (bounded commutative Rl-monoid). We introduce and study the ordinal sum (product) of two bounded BCK algebras.