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A note on pseudo-Kleene algebras. (English) Zbl 1431.06003

Summary: We introduce the concept of a pseudo-Kleene algebra which is a non-distributive modification of a Kleene algebra introduced by J. A. Kalman [Trans. Am. Math. Soc. 87, 485–491 (1958; Zbl 0228.06003)]. Basic properties of pseudo-Kleene algebras are studied. For pseudo-Kleene algebras with a fix-point there are determined subdirectly irreducible members.

MSC:

06D30 De Morgan algebras, Łukasiewicz algebras (lattice-theoretic aspects)
06B05 Structure theory of lattices
06D05 Structure and representation theory of distributive lattices

Citations:

Zbl 0228.06003
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References:

[1] Birkhoff, G.: Lattice Theory. Amer. Math. Soc. Colloqnium Publ. 25, second edition, American Mathematical Society, New York, 1948. | · Zbl 0033.10103
[2] Cignoli, R.: Injective de Morgan and Kleene algebras. Proc. Amer. Math. Soc. 47 (1975), 269-278. | | · Zbl 0301.06009
[3] Crvenkovič, S., Madarasz, R. Sz.: On Kleene algebras. Theoret. Comput. Sci. 108 (1993), 17-24. | | · Zbl 0778.03006
[4] Kalman, J. A.: Lattices with involution. Trans. Amer. Math. Soc. 87 (1958), 485-491. | | · Zbl 0228.06003
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