What question did this study set out to answer?

This research aims to establish the conditions under which formal ordinal sums of t-norms become t-norms on modular lattices.

March 25, 2026Open Access

Formal Ordinal Sums of Triangular Norms on Modular Lattices

Key Points

This research aims to establish the conditions under which formal ordinal sums of t-norms become t-norms on modular lattices.
Defined formal ordinal sums of t-norms in the context of bounded lattices.
Identified equivalent conditions to those proposed by Saminger.
Proved existence of the largest sublattice for non-overlapped subintervals.
Established necessary and sufficient conditions for formal ordinal sums to be t-norms.
Demonstrated a largest sublattice structure enabling t-norms within modular lattices.

Abstract

The formal ordinal sums of t-norms on the bounded lattice is defined by Saminger, but the formal ordinal sum need not be a t-norm. Hence, she provides a necessary and sufficient condition such that the formal ordinal sum is a t-norm indeed. In this paper, we first give several conditions that are equivalent to Saminger’s condition. Secondly, we prove that there is a largest sublattice containing all the pairwise non-overlapped subintervals such that the formal ordinal sum is a t-norm on this sublattice when the bounded lattice is modular.

Formal Ordinal Sums of Triangular Norms on Modular Lattices

Key Points

Abstract

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