Appendix A: Join, Meet, and Negation Operations for Non-Interval Type 2 Fuzzy Sets

A.l Introduction

General type 2 fuzzy sets (GT2FSs) are characterized by secondary memberships, which take any value between 0 and 1 (unlike interval type 2 fuzzy sets (IT2FSs), whose secondary memberships are either 0 or 1). The meet and join operations for GT2FSs, which represent the intersection and union for these sets, respectively, are supported by the extension principle by Zadeh as generalization of the intersection and union for type 1 fuzzy sets. In 2001, the initial work by Karnik and Mendel presented a simplified procedure to compute these operations for GT2FSs, although it trusted the condition that the secondary grades of type 2 fuzzy sets were normal and convex type 1 fuzzy sets. This work was later generalized by Coupland and John to include non-normal sets by borrowing some methods (Weiler-Atherton, modified Weiler Atherton, Bentley-Ottmann Plane Sweep Algorithm, etc.) from the sector of computational geometry; yet, convexity remained a necessary condition. Newer works studied the geometrical properties of some GT2FSs to seek out closed formulas or approximations for the join and meet operations in some specific cases.

Recent developments in type 2 symbolic logic have changed the perception researchers have of IT2FSs. IT2FSs are type 2 fuzzy sets in which uncertainty is equally distributed within the dimension (also called secondary membership), and thus, these secondary memberships are either

• 0 or 1, unlike GT2FSs, in which uncertainty within the dimension isn't equally weighted and, therefore, the distribution is often an arbitrary type
• 1 fuzzy set. When IT2FSs were initially defined, all the idea and operations were supported the precise case where IT2FSs are like interval-valued fuzzy sets (IVFSs). However, it's been recently shown that IT2FSs are more general than IVFSs. Hence, so as to derive the idea of those general sorts of interval type 2 symbolic logic systems (IT2FLSs) (which employ IT2FSs which aren't like IVFSs), it's necessary to develop the meet and join operations of GT2FSs with non-convex secondary memberships and, then, particularize it to the case of IT2FSs, which have secondary grades adequate to either 0 or 1. Hence, we'll be finding the join and meet operations for

GT2FSs where secondary memberships are arbitrary type 1 sets and, hence, are often non-convex and/or non-normal. This may be wont to derive the join and meet operations of IT2FSs where the secondary grades are non- convex sets.

T-norm fuzzy logics are a family of non-classical logics, informally delimited by having a semantics that takes the important unit interval [0,1] for the system of truth values and functions called t-norms for permissible interpretations of conjunction. They're mainly utilized in applied symbolic logic and fuzzy pure mathematics as a theoretical basis for approximate reasoning.

T-norm fuzzy logics belong in broader classes of fuzzy logics and manyvalued logics. So as to get a well-behaved implication, the t-norms are usually required to be left-continuous; logics of left-continuous t-norms further belong within the class of sub-structural logics, among which they're marked with the validity of the law of prelinearity, (A -> B) v (B -> A). Both propositional and first-order (or higher-order) t-norm fuzzy logics, as well as their expansions by modal and other operators, are studied. Logics that restrict the t-norm semantics to a subset of the important unit intervals (for example, finitely valued Lukasiewicz logics) are also usually included within the class.

Important samples of t-norm fuzzy logics are monoidal t-norm logic MTL of all left-continuous t-norms, basic logic BL of all continuous t-norms, product symbolic logic of the merchandise t-norm, or the nilpotent minimum logic of the nilpotent minimum t-norm. Some independently motivated logics belong among t-norm fuzzy logics, too; for instance, Lukasiewicz logic (which is that the logic of the Lukasiewicz t-norm) or Godel-Dummett logic (which is that the logic of the minimum t-norm).

In a partially ordered set P, the join and meet of a subset S are, respectively, the supremum (least upper bound) of S, denoted as vS, and infimum (greatest lower bound) of S, denoted as aS. Generally, the join and meet of a subset of a partially ordered set needn't exist; once they do exist, they're elements of P.

Join and meet can also be defined as a commutative, associative, and idempotent partial Boolean operation on pairs of elements from P. If a and b are elements from P, the join is denoted as a v b and, therefore, the meet is denoted as а л b.

Join and meet are symmetric duals with reference to order inversion. The join/meet of a subset of a completely ordered set is just its maximal/minimal element, if such a component exists.

A partially ordered set during which all pairs have a join may be a join- semilattice. Dually, a partially ordered set during which all pairs have a meet may be a meet-semilattice. A partially ordered set that's both a join-semilattice and a meet-semilattice may be a lattice. A lattice during which every subset, not just every pair, possesses a meet and a join may be a complete lattice. It's also possible to define a partial lattice, during which not all pairs have a meet or a join, but the operations (when defined) satisfy certain axioms.

А.2 Join

Join may be a lattice-theoretic concept that requires not to have anything to try to with unions. As an example, the positive integers partially ordered by divisibility are a lattice during which the join of two integers is their least integer. An example is R² partially ordered in order that

in that lattice

If X may be a set, and T is that the set of all topologies on X, (T, C) may be a lattice, the join of two topologies generally isn't their union: rather, it's the topology generated by taking their union as a sub-base.

A.3 Meet

Meet may be a lattice-theoretic concept that requires not to have anything to try to with intersection. As an example, the positive integers partially ordered by divisibility are a lattice during which the meet of two integers is their greatest common factor. An example is R² partially ordered in order that

in that lattice

A.4 Negation

The membership function of the complement of a fuzzy set A with membership function |iA is defined as the negation of the specified membership function. This is called the negation criterion. The Complement operation in fuzzy set theory is the equivalent of the NOT operation in Boolean algebra.