Construction of T-norms - Ordinal Sums

Ordinal Sums

The ordinal sum constructs a t-norm from a family of t-norms, by shrinking them into disjoint subintervals of the interval and completing the t-norm by using the minimum on the rest of the unit square. It is based on the following theorem:

Let T_i for i in an index set I be a family of t-norms and (a_i, b_i) a family of pairwise disjoint (non-empty) open subintervals of . Then the function T: 2 → defined as

is a t-norm.

The resulting t-norm is called the ordinal sum of the summands (T_i, a_i, b_i) for i in I, denoted by

or if I is finite.

Ordinal sums of t-norms enjoy the following properties:

• Each t-norm is a trivial ordinal sum of itself on the whole interval .

• The empty ordinal sum (for the empty index set) yields the minimum t-norm T_min. Summands with the minimum t-norm can arbitrarily be added or omitted without changing the resulting t-norm.

• It can be assumed without loss of generality that the index set is countable, since the real line can only contain at most countably many disjoint subintervals.

• An ordinal sum of t-norm is continuous if and only if each summand is a continuous t-norm. (Analogously for left-continuity.)

• An ordinal sum is Archimedean if and only if it is a trivial sum of one Archimedean t-norm on the whole unit interval.

• An ordinal sum has zero divisors if and only if for some index i, a_i = 0 and T_i has zero divisors. (Analogously for nilpotent elements.)

If is a left-continuous t-norm, then its residuum R is given as follows:

where R_i is the residuum of T_i, for each i in I.