Geodesic
Directoids with an antitone involution
Commentationes Mathematicae Universitatis Carolinae, Tome 48 (2007) no. 4, pp. 555-569 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
Classification : 06A06, 06A12, 06E20, 16Y99
Keywords: directoid; antitone involution; D-quasiring; symmetrical difference; direct decomposition 
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Chajda, I.; Kolařík, M. Directoids with an antitone involution. Commentationes Mathematicae Universitatis Carolinae, Tome 48 (2007) no. 4, pp. 555-569. http://geodesic.mathdoc.fr/item/CMUC_2007_48_4_a0/