## Directoids with sectionally switching involutions.(English)Zbl 1123.06001

The concept of directoid was introduced (by J. Ježek and R. Quackenbush) in order to axiomatize algebraic structures defined on upward directed ordered sets; in a certain sense, directoids generalize semilattices. A pair $$\mathcal{A}=(A;\sqcup)$$, where $$(A;\leq)$$ is an upward directed ordered set (which means that for every $$x,y\in A, U(x,y)=\{a\in A:x\leq a$$ and $$y\leq a\}\neq \emptyset$$) and $$\sqcup$$ denotes a binary operation on $$A$$, is called a directoid if: (i) $$x \sqcup y\in U(x,y)$$ for all $$x,y\in A$$; (ii) if $$x\leq y$$ then $$x\sqcup y=y$$ and $$y\sqcup x=y$$. In this paper the author shows that directoids with sectional switching involutions can be represented by weak d-implication algebras and, finally, that the correspondence between commutative directoids with 1 and with sectional switching involutions and d-algebras is one-to-one.

### MSC:

 06A11 Algebraic aspects of posets 06A12 Semilattices 03G25 Other algebras related to logic
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### References:

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