Note on orthocomplemented posets. (English) Zbl 0536.06002

Topology and measure III, Proc. Conf., Vitte/Hiddensee 1980, Part 1, 65-73 (1982).
[For the entire collection see Zbl 0499.00011.]
An orthocomplemented poset \({\mathcal P}=(P,\leq,0,1,^ \perp)\) is a poset with 0 and 1 with the operation of orthocomplementation such that \(a^{\perp \perp}=a, a\leq b\to b^{\perp}\leq a^{\perp}, a\vee a^{\perp}=1.\) Examples of orthocomplemented posets are given in the paper. One of them: the set of all left and right intervals of a given poset \((P,\leq,0,1)\) with inclusion (as order) and orthocomplement \(\{x:\quad x\leq a\}^{\perp}=\{x:\quad a\leq x\},\) has some importance in the second paper of the same author [Rend. Circ. Mat. Palermo, II. Ser. Suppl. 2, 67-74 (1982; Zbl 0535.06003)]. An orthocomplemented poset is called orthomodular if for every x,\(y\in P\) such that \(x\leq y\) there exists \(x^{\perp}\wedge y\) and \(x\vee(x^{\perp}\wedge y)=y.\) There is proved the theorem: For every ring \({\mathcal R}=(R,+,.,1)\) with unit the set \(Idem({\mathcal R})=\{x:\quad x^ 2=x\}\) is an orthocomplemented orthomodular poset with respect to \(x\leq y:\Leftrightarrow x\cdot y=y\cdot x=x\) and \(x^{\perp}:=1-x.\) This theorem extends a result of G. Birkhoff [Lattice theory (1967; Zbl 0153.025)] for rings with involution.
Reviewer: J.Waszkiewicz


06A06 Partial orders, general
06C15 Complemented lattices, orthocomplemented lattices and posets
16U99 Conditions on elements