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Directoids with an antitone involution. (English) Zbl 1199.06012
Summary: We investigate $$\sqcap$$-directoids which are bounded and equipped with a unary operation that is an antitone involution. Hence, a new operation $$\sqcup$$ can be introduced via De Morgan’s laws. Basic properties of these algebras are established. On every such 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.
Reviewer: Reviewer (Berlin)

##### MSC:
 06A12 Semilattices 06A06 Partial orders, general 06E20 Ring-theoretic properties of Boolean algebras 16Y99 Generalizations
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