It is NP-complete to determine whether a given ordered set has a fixed point free orderpreserving self-map. On the way to this result, we establish the NP-completeness of a related problem: Given ordered sets P and Q with t-tuples (PI,. ,pt) and (41,. . , qt) from P and Q respectively, is there an order-preserving map f: P + Q satisfying f(pz) > qr for each i = l,...,t? Mathematics Subject Clas...

‎Inspired by the work of Suzuki in‎ ‎[T. Suzuki‎, ‎A generalized Banach contraction principle that characterizes metric completeness‎, Proc‎. ‎Amer‎. ‎Math‎. ‎Soc. ‎136 (2008)‎, ‎1861--1869]‎, ‎we prove a fixed point theorem for contractive mappings‎ ‎that generalizes a theorem of Geraghty in [M.A‎. ‎Geraghty‎, ‎On contractive mappings‎, ‎Proc‎. ‎Amer‎. ‎Math‎. ‎Soc., ‎40 (1973)‎, ‎604--608]‎an...

We prove the following completeness theorem: If the fixed point operation over a category is defined by initial@, then the equations satisfied by the fixed point operation are exactly those of iteration theories. Thus, in such categories, the equational axioms of iteration theories provide a sound and complete axiomatization of the equational properties of the fixed point operation.

In this paper, we prove several fixed point theorems, which are generalizations of the Banach contraction principle and Kannan’s fixed point theorem. Further we discuss a characterization of metric completeness.

The description logic EL belongs to the family of logic-based knowledge representation formalisms. It allows a user to define concepts with the help of concept names (Nc), role names (Nr) and constructors: conjunction (u), existential restriction (∃r.C for r ∈ Nr and a concept C) and top constructor (>). Unification in Description Logics has been introduced in [6]. A unification problem in such...