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3) Idempotency: If env([H.sup.P.sub.i]) = env([H.sup.P])for all i and all of the [H.sup.P.sub.j] have the same probability information, i.e., [P.sub.i] = P, [for all]i, where P is the probabilistic weighting vector of [H.sup.P], then [H.sup.P.sub.i] = [H.sup.P] and [mathematical expression not reproducible] for all j, then
(1) Idempotency: if [[??].sub.i] = ([[a.sub.i],[b.sub.i],[c.sub.i]], [[d.sub.i],[e.sub.i],[f.sub.i]]) = [??] = ([a,b,c], [d,e,f]), for all i, then
(P5) Idempotency: assume that [mathematical expression not reproducible] for all i [member of] [n].
(2) Idempotency: if ([s.sub.i], 0) = (s, 0) for all u = 1, ..., n, then
Pin, "Tropical semirings," in Idempotency (Bristol, 1994), vol.
--the square of the matrix coincides with the matrix itself (the idempotency property)
[therefore] h([I.sub.1])[union]([I.sub.2])) [subset or equal to] h([I.sub.1] [union] [I.sub.2]) (Idempotency)
Bootstrapping [8, Theorem 4.1] by use of idempotency of projections, we obtain the following result, which will allow us to control the error incurred by approximating the Neumann data.
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.
In this subsection, we consider algebraic properties regarding idempotency of soft product operations and show that IVF soft sets have some nonclassical algebraic properties, compared with interval-valued fuzzy sets.
Brummer, Natural extensions of the [T.sub.0]-spaces, their idempotency, and the quasi-uniform bicompletion, Sum Topo 2001, Sixteenth Summer Conference on Topology and its Applications, July, 18-21,2001.
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