Hypergroupoids on Partially Ordered Sets Josef Zapletal

A hypergroupoid (or a multigroupoid) is a pair (M, ◦) where M is a nonempty set and ◦ : M × M → P∗(M) is a binary hyperoperation also called a multioperation. P∗(M) is the system of all nonepmty subsets of M. Partially ordered set M with the ordering ≤ with the greatest element I is in this article denoted withM = (M,≤, I). On M = (M,≤, I) for arbitrary x, y ∈M, we define a binary hyperoperation ◦ as follows: x ◦ y = {min(X ∩ Y)}. where X = {xi | xi ≥ x} and Y = {yi | yi ≥ y} We then denote the set M with the defined binary operation with M = (M ≤, ◦, I). It is proved that the hyperoperation ◦ on (M = (M ≤, ◦, I) is idempotent and commutative but not associative. Hence the partially ordered set M with the operation ◦ is a commutative hypergroupoid. In the beginning of the second chapter a definition of congruence on a commutative hypergroupoid M is given. By a congruence we call a relation of equivalence ρ on M such that for every quadruple of elements a1, a2, b1, b2 ∈ M for which a1ρ b1, a2ρ b2 the following holds: For every x ∈ a1 ◦a2 there exists y ∈ b1 ◦b2 and for every y ′ ∈ b1 ◦b2 there exists x ′ ∈ a1 ◦ a2 with the property xρ y and x′ρ y′. See [4] p.151 and [10]. It is shown that the relation of substitutabality Ξ(M,L) satisfies this definition. The distinguishing subsets of commutative hypergroupoids are studied in the third chapter. The fourth chapter contains a concrete partially ordered set Q (Figure 1) with the hyperoperation ◦. The congruence Ξ(Q,L) with its partition of the set Q is given as an example. The other example demonstrates a distinguishing subset of Q. Partially ordered set