Orthomodular lattices as L -algebras
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Orthomodular lattices as L-algebras Yali Wu1,2 · Yichuan Yang1 Published online: 8 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We first prove that the axioms system of orthomodular L-algebra (O-L-algebras for short) as given in [Rump: Forum Mathematicum, 30(4), 2018: 973–995] are not independent by giving an independent axiom one. Then, two conditions for K L-algebras to be Boolean are provided. Furthermore, some theorems of Holland are reproved using the self-similar closure of O M-L-algebras. In particular, the monoid operation of the self-similar closure is shown to be commutative. Keywords Orthomodular lattice · L-algebra · Self-similar closure · O M-L-algebra
1 Introduction Orthomodular lattices (O M Ls for short) play an important role in the mathematical foundations of quantum theories as so-called quantum logics. In comparison with the classical model, a Boolean algebra, the distributive law is replaced by the orthomodular one. Orthomodularity can be considered as the smallest weakening of distributivity which is satisfied by both the classical model (a Boolean algebra) and the standard model of quantum mechanics (the projection lattice of a Hilbert space Birkhoff and Neumann 1936). Orthomodular lattices have been investigated by many authors, including Kalmbach (1983), Greechie (1968, 1977), Günter (1979), Beran (1985), etc. L-algebras are related to algebraic logics and quantum structures. They were introduced by Rump (2008). Hilbert algebras, locales, (left) hoops, (pseudo) MV-algebras, and l-group cones, are L-algebras. In Rump and Yang (2012), showed that an L-algebra is representable as an interval in a lattice-ordered group if and only if it is semiregular with a smallest element and a bijective negation. They Communicated by A. Di Nola.
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Yichuan Yang [email protected] Yali Wu [email protected]
1
School of Mathematical Sciences, Beihang University, Shahe Campus, Beijing 102206, China
2
School of Mathematics and Physics, Hebei GEO University, Shijiazhuang 050031, China
also characterized pseudo-MV algebras and Bosbach’s noncommutative bricks as L-algebras (Yang and Rump 2012). In the present paper, we give a sufficient and necessary condition when O M-L-algebras are orthomodular lattices (Theorem 2), which shows that the axioms system of orthomodular L-algebras given in Rump (2018) are not independent. As a special case, two representations of a Boolean algebra as a K L-algebra are obtained (Theorems 3, 4). Furthermore, the self-similar closure of an O M-L-algebra is used to give a new interpretation of the commutativity relation aCb (i.e., a = (a∧b)∨(a∧b )). We show that it coincides with commutation in algebras, i.e., aCb ⇔ ab = ba. As a consequence, we obtain new proofs of some results in Holland (1963, P. 68-71, Theorems 2,3,4,5).
2 Preliminaries Recall that an orthomodular lattice (O M L) is a structure (L, ≤, , 0, 1), where (L, ≤, 0, 1) is a lattice with minimum 0 and maximum 1, is a unary operation on L such that the following condi
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