Antitone involution; De Morgan laws; Kleene algebra; distributive lattice; pseudo-Kleene algebra; subdirectly irreducible
We introduce the concept of a pseudo-Kleene algebra which is a non-distributive modification of a Kleene algebra introduced by J. A. Kalman [Kalman, J. A.: Lattices with involution. Trans. Amer. Math. Soc. 87 (1958), 485–491.]. Basic properties of pseudo-Kleene algebras are studied. For pseudo-Kleene algebras with a fix-point there are determined subdirectly irreducible members.
 Birkhoff, G.: Lattice Theory
. Amer. Math. Soc. Colloqnium Publ. 25, second edition, American Mathematical Society, New York, 1948. MR 0029876
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