# zbMATH — the first resource for mathematics

On the axioms of multistructures. (Sur les axiomes des multistructures.) (Russian. French summary) Zbl 0075.01901
Let $$M$$ be a nonempty set and let $$x\lor y$$, $$x \wedge y$$ be two maps from $$M\times M$$ into the set of all subsets of $$M$$. For every $$A,B\subset M$$, we put: $$A\lor B = \bigcup_{a\in A, b\in B} (a\lor b)$$, $$A\wedge B = \bigcup_{a\in A, b\in B}(a \wedge b)$$, if $$A\neq \emptyset\neq B$$, and $$A\lor B = A\wedge B = \emptyset$$, if any of the sets $$A,B$$ is empty. The dual of a proposition concerning $$M,\lor, \wedge$$, is defined in a natural way. The set $$M$$ is said to be a multilattice, if the following axioms are satisfied:
$$(M_1)$$ $$a\lor b = b \lor a$$, and dually;
$$(M_2)$$ if $$x\in (a \lor b)\lor c$$, then there exists $$x'\in a\lor (b\lor c)$$, such that $$x\lor x' = \{x\}$$, and dually;
$$(M_3)$$ if $$a\lor b\neq 0$$, then $$a \wedge (a\lor b) = a$$, and dually;
(M4) $$a\lor a\neq \emptyset$$, and dually;
$$(M_5)$$ if $$a = b$$, then $$a\lor c = b\lor c$$, and dually; $$(M_6)$$ if $$x,x\in a\lor b$$, $$x^*\in x\lor x'$$, $$x\neq x'$$, then $$x\neq x^*\neq x'$$, and dually. A multilattice is called associative, if $$(a\lor b)\lor c = a\lor (b\lor c)$$ and dually. These notions were introduced by M. Benado [Czech. Math. 5, 308–344 (1955; Zbl 0075.01803)] who asked the question whether $$(M_6)$$ is independent from $$(M_1)-(M_5)$$ and whether there exist associative multilattices that are not lattices. The present author proves that the answer to both problems is affirmative.

##### MSC:
 06B05 Structure theory of lattices
multistructures
Full Text: