Triangular Norms

Synonyms

t-Norm

Definition

Triangular norms (briefly t-norms) are special binary operations T: [0, 1]² → [0, 1]. They are interesting for fuzzy logic because they preserve the fundamental properties of the logical conjunction “and” (to hold at the same time), namely commutativity, monotonicity, associativity, and boundedness and thus, they serve as a natural generalization of the classical conjunction in many-valued logical systems.

A concept associated with the t-norm is the triangular conorm (t-conorm) S: [0, 1]² → [0, 1]. This corresponds to the behaviour of truth values when joined by the logical connective “or.”

Key Points

A t-norm is a binary operation T: [0, 1]² → [0, 1] such that the following axioms are satisfied for all a, b, c ∈ [0, 1]:

$$ {\begin{array}{l}\left(\mathrm{commutativity}\right)\qquad \quad a\ \mathbf{T}\ b=b\ \mathbf{T}\ a,\\ {}\left(\mathrm{a}\mathrm{ssociativity}\right)\quad \mathrm{a}\ \mathbf{T}\left(\mathrm{b}\ \mathbf{T}\...