Geometry of left-continuous triangular forms with strong induced negations

The purpose of this paper is to make the first step toward the understanding of the structure of left-continuous triangular norms with strong induced negations. For two-placed functions on [0,1] two properties are introduced: The rotation invariance property and the self quasi-inverse property. It is proved that these properties are characteristic for the class left-continuous triangular norms with strong induced negations. The two properties turn out to be equivalent on the class of symmetric, non-decreasing two-place functions on [0,1], that is, such a function admits the rotation invariance property if and only if it admits the self quasi-inverse property. These properties have equivalent geometrical counterparts which are investigated, explained in detail and examples are given. These geometrical counterparts can be represented in 3 and 2 dimensions.