Non-preservation of chosen properties of fuzzy relational compositions based on fuzzy quantifiers

Fuzzy relational compositions based on fuzzy quantifiers naturally do not preserve all the properties that are preserved for “standard” fuzzy relational compositions and, in many cases, the property is preserved only in a weaker form. For example, the associativity, that is preserved in the standard case derived from the universal and the existential quantifiers, generally does not hold for the case of compositions based on fuzzy quantifiers. However, is it the case that only the standard quantifiers lead to the preservation of such properties? Without any restriction on the shape of the fuzzy relations, the answer is positive.

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