## I. Chajda, M. Kola\v {r}\'{\i }k

*Directoids with an antitone involution *

Comment.Math.Univ.Carolin. 48,4 (2007) 555-569. **Abstract:**We investigate $\sqcap $-directoids which are bounded and equipped by a unary operation which is an antitone involution. Hence, a new operation $\sqcup $ can be introduced via De Morgan laws. Basic properties of these algebras are established. On every such an algebra a ring-like structure can be derived whose axioms are similar to that of a generalized boolean quasiring. We introduce a concept of symmetrical difference and prove its basic properties. Finally, we study conditions of direct decomposability of directoids with an antitone involution.

**Keywords:** directoid, antitone involution, D-quasiring, symmetrical difference, direct decomposition

**AMS Subject Classification:** 06A12, 06A06, 06E20, 16Y99