An operator connection is a binary operation assigned to each pair of positive operators satisfying monotonicity, continuity from above, and the transformer inequality.
An operator mean is a binary operation σ defined on the set of strictly positive operators, if the following conditions hold: begin{aligned}& (1 quad Aleq C,Bleq D quadRightarrowquad Asigma Bleq Csigma D. & (2)quad A_{n}downarrow A,B_{n} downarrow B quad Rightarrowquad A_{n}sigma B_{n}downarrow Asigma B. & (3)quad T^(Asigma B) Tleqbigl(T^ATbigr)sigmabigl(T^BTbigr)quad text{for } Tin B H).
where ∗ is a binary operation on X.
A triangular norm T (t-norm for short) is a binary operation on L which is commutative, associative, monotone and has a neutral element 1.
A triangle function τ is a binary operation on D+, which is commutative, associative, and τ f,H0)=f for every f∈D+.
A triangle function τ is a binary operation on D +, which is commutative, associative and τ ( f, H 0 ) = f for every f ∈ D +.
Let us recall [3] that a continuous t-norm is a binary operation ∗ : [ 0, 1 ] × [ 0, 1 ] → [ 0, 1 ] such that ( [ 0, 1 ], ≤, ∗ ) is an ordered Abelian topological monoid with unit 1.
The most general circumstance we shall deal with is the situation where (X, ∗) is actually a groupoid, i.e., the product operation ∗ is a binary operation, where we assume no restrictions a priori.
The most general circumstance we will deal with in this paper is the situation where ( X, ∗ ) is actually a groupoid, i.e., the product operation ∗ is a binary operation, where we assume no restrictions a priori.
