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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
Keywords:
multistructures
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