Aggregation on Boolean multi-polar space: Knowledge-based vs. category-based ordering

Three types of aggregation operators defined on a Boolean multi-polar space {0,1,…,m}{0,1,…,m} are compared. Three different monotonicity conditions that yield different types of aggregation operators are discussed and the structure of these aggregation operators on {0,1,…,m}{0,1,…,m} with additional properties of associativity and commutativity is studied for any m∈Nm∈N. The commutative associative Boolean multi-polar aggregation operators are shown to coincide with Boolean multi-polar uninorms. The structure of commutative associative SL aggregation operators is shown to be dependent on the structure of the class of Abelian idempotent semigroups based on m+1m+1 elements with an annihilator. We show that each commutative associative SL aggregation operator with annihilator 0 corresponds to a lower semi-lattice with bottom element 0. For m>2m>2 the class of commutative associative C aggregation operators is proved to be equal to the intersection of the class of commutative associative Boolean multi-polar aggregation operators and the class of commutative associative SL aggregation operators. Examples of the three types of commutative associative aggregation operators are shown and discussed for m=1,2,3m=1,2,3.