Pseudosemirings induced by ortholattices. (English) Zbl 0879.06003

Chajda, Ivan

Czech. Math. J. 46, No. 3, 405-411 (1996).

An algebra \((A;+,\cdot ,0,1)\) is called a pseudosemiring if the operation \(\cdot\) is associative, if the operation \(+\) is commutative and if the following axioms are satisfied: (i) \(x+0=x=x\cdot 1=1\cdot x=x\), \(x\cdot 0=0\cdot x;\) (ii) \(x+(1+y)=(x+1)+y;\) (iii) \(\forall x\in A\;\exists y\in A\;x+y=0;\) (iv) \((1+xy)x=x+xyx\). A pseudoring is said to be commutative/idempotent if the operation \(\cdot\) is commutative/idempotent. A pseudoring is called an orthopseudoring if it is commutative, idempotent and if it satisfies the following postulates: (a) \(x+x=0;\) (b) \((1+x)(1+xy)=1+x\); (c) \(\big [1+x(1+y)\big ]\big [1+y(1+x)\big ]=1+(x+y)\). An orthopseudoring satisfying the axiom (d) \((x+xy)+xy=x\) is called an orthomodular pseudoring. Typical results: Thm 1. Let \((L;\vee ,\wedge ,{}^\perp ,0,1)\) be an ortholattice, let \(x+y:=(x\wedge y^\perp)\vee (x^\perp \wedge y)\) and \(x\cdot y:= x\wedge y\). Then \((L;+,\cdot ,0,1)\) is an orthopseudoring, called an orthopseudoring induced by \(L\). It is an orthomodular pseudoring, provided \(L\) is an orthomodular lattice. Thm 3. Let \(L\) be an ortholattice, \(P(L)\) the induced orthopseudoring and \(L\big (P(L)\big)\) the ortholattice induced by \(P(L)\). Then \(L=L\big (P(L)\big)\). Let \(P\) be an orthopseudoring, \(L(P)\) the induced ortholattice and \(P\big (L(P)\big)\) the orthopseudoring induced by \(L(P)\). Then \(P=P\big (L(P)\big)\).
The paper contains also some instructive examples completing the recorded assertions.

Reviewer: L.Beran (Praha)

MSC:

06C15

Complemented lattices, orthocomplemented lattices and posets

Keywords:

Boolean ring; Boolean algebra; ortholattice; orthomodular lattice; pseudosemiring

References:

[1]	Birkhoff G.: Lattice Theory, \(3^{\mathrm rd}\) edition. Publ. AMS, Providence, 1967.
[2]	Chajda I., Kotrle M.: Boolean semirings. Czech. Math. J. 44(119) (19941994), 763-767. · Zbl 0820.06008
[3]	Beran L.: Orthomodular Lattices, Algebraic Approach. Academia Praha, 1984. · Zbl 0558.06008