CONFIGURATIONAL STRUCTURE AND IDEALS OF BCC/WEAK BCC-ALGEBRAS

Abstract

The study of algebras motivated by known logics is interesting and very useful for corresponding logics. In such kinds of algebras we can introduce a natural partial order which has many interesting properties and gives decomposition of algebras into subsets called branches. In the theory of these algebras an important role plays also ideals and congruences. This thesis is devoted to study of branches and ideals of weak BCC- algebras called in China – BZ-algebras. Such algebras are strongly con- nected with a weak BIK+ -logic. Basic properties of branches and connec- tions between various types of ideals are described by using the so-called Dudek’s map. A new type of ideals are introduced and characterized. Con- nections between ideals saving some basic properties are investigated. De- compositions of weak BCC-algebras into branches are presented and it is proved that a decomposition of a weak BCC-algebra X into branches in- duces on X a congruence θ such that the quotient algebra X/θ is isomorphic to subalgebra of X containing only initial elements of X. A decomposition of a medial weak BCC-algebra into direct product of subalgebras generated by one element is presented too. Next we propose two new decompositions which are in fact two new generalizations of weak BCC-algebras. The first decomposition is a decomposition of a basic set X into a set theoretic union of two weak BCC-algebras X1 and X2 having common distinguished ele- ment. The second decomposition is a decomposition of a basic operation defined on X into two new operations in this way that these two operations save a distinguished element of X, a partial order defined on X and some basic properties of weak BCC-algebras.