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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:

[1] Abbott J. C.: Semi-boolean algebras. Matem. Vestnik 4 (1967), 177-198. · Zbl 0153.02704
[2] Chajda I.: Lattices and semilattices having an antitone involution in every upper interval. Comment. Math. Univ. Carol. 44 (2003), 577-585. · Zbl 1101.06003
[3] Chajda I., Halaš R., Kühr J.: Distributive lattices with sectionally antitone involutions. Acta Sci. Math. (Szeged), 71 (2005), 19-33. · Zbl 1099.06006
[4] Ježek J., Quackenbush R.: Directoids: algebraic models of up-directed sets. Algebra Universalis 27 (1990), 49-69. · Zbl 0699.08002 · doi:10.1007/BF01190253
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