Note on a More General Setting for Complemented Elements (I) (Remarks from Semirings Theory)
Several applications, e.g., fuzzy modelling or the programs semantics, motivate an interest for the study of the set of complemented elements of an algebra with two composition laws. In this part of the note one considers an algebra S = (S; +, ·, 0, 1) (Definition 2), C (S) being its subset of complemented elements. S is assumed to have the multiplication distributive over the addition, 0 as neutral element and 1 as a right identity for the elements of C (S) and for their complements and such that 01 = 10 = 0. One obtains necessary and sufficient conditions for organizing C (S) as a Boolean Algebra. Within the more general setting provided by S one thus extends some results given in the monograph of Golan (1992) for the particular case of semirings.