Comparability Axioms in Orthomodular Lattices and Rings with Involution
In this article, a Schr\({\ddot{\mathrm{o}}}\) der–Bernstein type theorem is proved for orthomodular lattices. Various comparability axioms available in Baer \(*\) -rings are introduced in orthomodular lattices. Some applications to complete orthomodular
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Abstract In this article, a Schr¨oder–Bernstein type theorem is proved for orthomodular lattices. Various comparability axioms available in Baer ∗-rings are introduced in orthomodular lattices. Some applications to complete orthomodular lattices are given. The related classical results in Baer ∗-rings are generalized to ∗-rings. Keywords Orthomodular lattice · Psuedocomplemented lattice · Relatively semiorthocomplemented lattice · Parallelogram law MSC(2010) Primary: 06C15 · Secondary: 06D15
1 Introduction We assume that the reader is familiar with basics of lattice theory. A bounded lattice is an algebra (L , (∧, ∨, 0, 1)) where (L , ∧, ∨) is a lattice with 0 and 1. Two elements a and b of a lattice L are said to form a modular pair, denoted by (a, b)M, when (c ∨ a) ∧ b = c ∨ (a ∧ b) holds for all c ≤ b. An element z of a lattice L with 0 and 1 is called a central element when there exist two lattices L 1 and L 2 and an isomorphism between L and the direct product of L 1 , L 2 such that z corresponds to the element [11 , 02 ] ∈ L 1 × L 2 . The set of all central elements of L is called the center of L and it is denoted by Z (L). An orthocomplementation on a bounded lattice N.K. Thakare 13, Nimbapushpa, Jawahar Nagar, O Sakri Road, Dhule 424001, India e-mail: [email protected] B.N. Waphare (B) Center for Advanced Study in Mathematics, Savitribai Phule Pune University, Pune 411007, India e-mail: [email protected]; [email protected] A. Patil Garware College of Commerce, Karve Road, Pune 411004, India e-mail: [email protected] © Springer Science+Business Media Singapore 2016 S.T. Rizvi et al. (eds.), Algebra and its Applications, Springer Proceedings in Mathematics & Statistics 174, DOI 10.1007/978-981-10-1651-6_13
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is a unary operation satisfying a ∨ a ⊥ = 1, a ∧ a ⊥ = 0, a ≤ b implies b⊥ ≤ a ⊥ , (a ⊥ )⊥ = a. An easy consequence of this are DeMorgan laws (a ∨ b)⊥ = a ⊥ ∧ b⊥ , (a ∧ b)⊥ = a ⊥ ∨ b⊥ . An ortholattice is an algebra (L , (∧, ∨,⊥ , 0, 1)) where (L , (∧, ∨, 0, 1)) is a bounded lattice and ⊥ is an orthocomplementation on it. An orthomodular lattice (abbreviated: OML) is an ortholattice satisfying the orthomodular law: ‘If a ≤ b, then a ∨ (a ⊥ ∧ b) = b’. This law can be again replaced by the equation ‘a ∨ (a ⊥ ∧ (a ∨ b)) = a ∨ b’, see Kalmbach [4] and Stern [11]. Two elements a and b of an OML are said to be strong perspective if they have a common complement in [0, a ∨ b]. The relative center property holds in an OML L, if the center of any interval [0, a] of L is the set {a ∧ c | c ∈ Z (L)}. In the second section, we consider a relatively semi-orthocomplemented lattice L with 0 and 1 and an equivalence relation on L satisfying some conditions. A Schr¨oder– Bernstein type theorem is proved for OMLs. Similar results were proved in [7, 9]. Here we release the assumptions namely, orthogonal additivity and completeness in OMLs. We introduce comparability axioms and finiteness in OMLs. In Baer ∗-rings, several comparability axioms such as parallelogram law, generalized c
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