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Pseudosemirings induced by ortholattices. (English) Zbl 0879.06003
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
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##### References:
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