## 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

### MSC:

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

### Keywords:

orthocomplemented posets; orthomodular poset

### Citations:

Zbl 0499.00011; Zbl 0535.06003; Zbl 0153.025