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Commutative hypergroups in the sense of Marty and ordered sets. (English) Zbl 0827.20085
Chajda, I. (ed.) et al., General algebra and ordered sets. Proceedings of the international conference and summer school, held in Horní Lipová, Czech Republic, September 4-12, 1994. Olomouc: Palacký University Olomouc, Department of Algebra and Geometry, 19-30 (1994).
The paper deals with hyperstructures which are defined on ordered sets. The object is investigated from the category theory point of view. A full subcategory $${\mathbf K}$$ of the category AHG (of all commutative hypergroups and their homomorphisms) is defined and it is proved that the functors $$F_+, F_- : \text{\textbf{P}os} \to {\mathbf K}$$ are isomorphisms and they are realizations. A necessary and sufficient condition in order that the upper and lower order hypergroups defined on an ordered set would be join spaces is proved. Finally, three equivalent conditions on an autonomous automaton are proved.
For the entire collection see [Zbl 0815.00007].

##### MSC:
 20N20 Hypergroups 06A06 Partial orders, general 18A22 Special properties of functors (faithful, full, etc.) 06F05 Ordered semigroups and monoids 20L05 Groupoids (i.e. small categories in which all morphisms are isomorphisms) 20M35 Semigroups in automata theory, linguistics, etc.