Basic algebras and L-algebras
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FOUNDATIONS
Basic algebras and L-algebras Jing Wang1 · Yali Wu2 · Yichuan Yang1
© Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this paper, we study the relation between L-algebras and basic algebras. In particular, we construct a lattice-ordered effect algebra which improves an example of Chajda et al. (Algebra Univ 60(1), 63–90, 2009). Keywords Basic algebras · L-algebras · MV-algebras · Orthomodular lattices · Effect algebras
1 Introduction Basic algebras, which generalize both MV-algebras and orthomodular lattices, were introduced in Chajda et al. (2009) and Chajda et al. (2007) as a common base for axiomatization of many-valued propositional logics as well as of the logic of quantum mechanics. The relationship between basic algebras, MV-algebras, orthomodular lattices and lattice-ordered effect algebras was considered in Botur (2010), Botur and Halaš (2008), Chajda (2012; 2015), Chajda et al. (2009). One can mention that every MV-algebra is a basic algebra whose induced lattice is distributive (Chajda 2015, P. 18, Lemma 5.2). The sufficient and necessary condition for an orthomodular lattice to be a basic algebra has been obtained in Chajda (2015, P. 17, Theorem 4.3). Relation between lattice-ordered effect algebras and basic algebras was treated in Botur and Halaš (2008), Chajda (2012) by considering their common lattice structure (a lattice with section antitone involutions).
Communicated by A. Di Nola.
L-algebras, which are related to algebraic logic and quantum structures, were introduced by Rump (2008). Many examples shown that L-algebras are very useful. Yang and Rump (2012), characterized pseudo-MV-algebras and Bosbach’s non-commutative bricks as L-algebras. Wu and Yang (2020) proved that orthomodular lattices form a special class of L-algebras in different ways. It was shown that every lattice-ordered effect algebra has an underlying L-algebra structure in Wu et al. (2019). In the present paper, we study the relationship between basic algebras and L-algebras. We prove that a basic algebra which satisfies (z ⊕ ¬x) ⊕ ¬(y ⊕ ¬x) = (z ⊕ ¬y) ⊕ ¬(x ⊕ ¬y) can be converted into an L-algebra (Theorem 1). Conversely, if an L-algebra with 0 and relation given by (10) such that it is an involutive bounded lattice can be organized into a basic algebra, it must be a lattice-ordered effect algebra (Theorem 2). Finally, we construct a lattice-ordered effect algebra which improves (Chajda et al. 2009, P. 80, Example 5.3).
Supported by CNNSF (Grant: 11771004).
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Yichuan Yang [email protected] Jing Wang [email protected] Yali Wu [email protected]
2 Preliminaries Note that basic algebras were introduced in Chajda (2007; 2009), but the axiomatic system was extended by one more axiom which is dependent on the following axioms as shown in Chajda and Kolšík (2009).
1
School of Mathematical and Sciences, Beihang University, Beijing 100191, China
Definition 1 A basic algebra is an algebra B = (B; ⊕, ¬, 0) of type (2, 1, 0) satisfying the following identities:
2
School of Mathematics and
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