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Tense operators on basic algebras. (English) Zbl 1246.81014
Effect algebras were introduced ca. 20 years ago to generalize the set of all Hermitian operators between zero and the identity operator of a Hilbert space. They are simply partial algebras with a primary operation $$+$$ of addition. They cannot be studied completely using methods of universal algebras. Therefore, the first three authors introduced basic algebras [Soft Comput. 14, No. 3, 251–255 (2010; Zbl 1188.03048)]. The aim of the present paper is to introduce tense operators for basic algebras and to study their basic properties. They have also connection with the so-called dynamic effect algebras.

##### MSC:
 81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects) 03G25 Other algebras related to logic 06F35 BCK-algebras, BCI-algebras
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##### References:
 [1] Botur, M.: An example of a commutative basic algebra which is not an MV-algebra. Math. Slovaca 60(2), 171–178 (2010) · Zbl 1240.06042 [2] Botur, M., Chajda, I., Halaš, R.: Are basic algebras residuated structures? Soft Comput. 14(3), 251–255 (2009) · Zbl 1188.03048 [3] Botur, M., Halaš, R.: Commutative basic algebras and non-associative fuzzy logics. Arch. Math. Log. 48(3–4), 243–255 (2009) · Zbl 1168.03014 [4] Botur, M., Halaš, R.: Complete commutative basic algebras. Order 24, 89–105 (2007) · Zbl 1128.06004 [5] Botur, M., Halaš, R.: Finite commutative basic algebras are MV-algebras. J. Mult.-Valued Log. Soft Comput. 14(1–2), 69–80 (2008) · Zbl 1236.06007 [6] Burges, J.: Basic tense logic. In: Gabbay, D.M., Günther, F. (eds.) Handbook of Philosophical Logic, vol. II, pp. 89–139. Reidel, Dordrecht (1984) [7] Chajda, I., Halaš, R., Kühr, J.: Semilattice Structures, Heldermann, Lemgo (2007). ISBN:978-3-88538-230-0 [8] Chajda, I., Halaš, R., Kühr, J.: Many-valued quantum algebras. Algebra Univers. 60(1), 63–90 (2009) · Zbl 1219.06013 [9] Chajda, I., Kolařík, M.: Dynamic effect algebras. Math. Slovaca (submitted) [10] Chajda, I., Kolařík, M.: Independence of axiom system of basic algebras. Soft Comput. 13(1), 41–43 (2009) · Zbl 1178.06007 [11] Diaconescu, D., Georgescu, G.: Tense operators on MV-algebras and Łukasiewicz-Moisil algebras. Fundam. Inform. 81, 1–30 (2007) · Zbl 1136.03045
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