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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.

Reviewer: Florentina Chirteş (Craiova)

### Keywords:

directoid; commutative directoid; semilattice; involution; implication algebra; sectionally switching mapping
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\textit{I. Chajda}, Acta Univ. Palacki. Olomuc., Fac. Rerum Nat., Math. 45, 35--41 (2006; Zbl 1123.06001)

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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 |

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