Turkish Journal of Mathematics

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On hypersemigroups




This is from the paper "Hypergroupes canoniques values et hypervalues" by J. Mittas in Mathematica Balkanica 1971: "The concept of hypergroup introduced by Fr. MARTY in 1934 [Actes du Congres des Math. Scand. Stocholm 1935, p. 45] is as follows: "A hypergroup is a nonempty set $H$ endowed with a multiplication $xy$ such that, for every $x,y,z\in H,$ the following hold: (1) $xy\subseteq H$; (2) $x(yz)=(xy)z$ and (3) $xH=Hx=H$. The first condition expresses that the multiplication is an hyperoperation on $H$, in other words, the composition of two elements $x,y$ of $H$ is a subset of $H$. It is very easy to prove that for any $x,y\in H$, we have $xy\not=\emptyset$." Although according to Mittas "it is very easy to prove that $xy\not=\emptyset$", this is not possible. The notation $x(yz)$ has a meaning of course if we identify the $x$ by $\{x\}$ and define an operation between sets. The authors working on hypersemigroups added in the definition by Mittas, the following: $x(yz)=(xy)z$ means that $\bigcup\limits_{u \in yz} {xu} = \bigcup\limits_{v \in xy} {vz}$. But we never use this last equality in the papers on hypersemigroups in which we always use the $x(yz)=(xy)z$. As a result, most of the results of ordered hypersemigroups are copies from corresponding results on ordered semigroups in which the multiplication "$\cdot$" has been replaced by "$\circ$".

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