Another important mathematical understanding that arrays can help to develop is that of

commutativity (Figure 3).

Based on these intuitionistic fuzzy Einstein operations and fuzzy measure, we develop some new aggregation operators, such as intuitionistic fuzzy Einstein Choquet averaging (IFCAS) operator, intuitionistic fuzzy Einstein Choquet geometric (IFCGS) operator, and study various special cases of the operators, and also investigate some desired properties of the developed operators, such as

commutativity, idempotency, boundary, etc.

Wei (2010a) investigated the multiple attribute group decision making (MAGDM) problems in which both the attribute weights and the expert weights take the form of real numbers, attribute values take the form of intuitionisticfuzzy numbers or interval-valued intuitionisticfuzzy numbers and proposed two new aggregation perators: induced intuitionistic fuzzy ordered weighted geometric (I-IFOWG) operator and induced interval-valued intuitionistic fuzzy ordered weighted geometric (I-IIFOWG) operator, and studied some desirable properties of the I-IFOWG and I-IIFOWG operators, such as

commutativity, idempotency and monotonicity.

Moreover, several authors studied

commutativity in prime and semiprime rings admitting derivations and generalized derivations which satisfy appropriate algebraic conditions on suitable subsets of the rings.

Kumar: Minimal

commutativity and common fixed points, J.

They include discussions of the Galois map and its induced maps, when direct sums of modules inherit certain properties, prime rings with left derivations, some

commutativity theorems concerning additive mappings and derivations on semi-prime rings, a short proof that continuous modules are clean, and imprimitive regular action in the ring of integers modulo n.

If we consider heaps up to poset isomorphism which preserve the labeling, then heaps encode precisely

commutativity classes, that is, if the word s' is obtainable from s by transposing commutating generators then there exists a poset isomorphism between [H.

Interestingly, the Intlab intersect operator does not fulfill a

commutativity relation of the kind intersect(a, b) = intersect([bar.

It is well known that

commutativity of matrices is very important in the theory of matrices.

As one can notice, the same operand role is used for both x and y to preserve

commutativity of multiplication and addition.

As pointed out in Section 3, the

commutativity between percentiles and monotonic transformations means that the log of a percentile ratio of earnings is equal to the difference between the two percentiles of log earnings, i.

Filippis, On derivations and

commutativity in prime rings, Int.