Geodesic
Commutative directoids with sectional involutions
Discussiones Mathematicae. General Algebra and Applications, Tome 27 (2007) no. 1, pp. 49-58.

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The concept of a commutative directoid was introduced by J. Ježek and R. Quackenbush in 1990. We complete this algebra with involutions in its sections and show that it can be converted into a certain implication algebra. Asking several additional conditions, we show whether this directoid is sectionally complemented or whether the section is an NMV-algebra.
Keywords: commutative directoid, sectional involution, sectional complement, d-implication algebra, NMV-algebra 
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Chajda, Ivan. Commutative directoids with sectional involutions. Discussiones Mathematicae. General Algebra and Applications, Tome 27 (2007) no. 1, pp. 49-58. http://geodesic.mathdoc.fr/item/DMGAA_2007_27_1_a2/

[1] I. Chajda and J. Kühr, A non-associative generalization of MV-algebras, Math. Slovaca, to appear.

[2] J. Ježek and R. Quackenbush, Directoids: algebraic models of up-directed sets, Algebra Universalis 27 (1990), 49-69.