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.


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