# Introduction

Chapter

## Abstract

Residuation is one of the most important concepts of the theory of ordered algebraic structures which naturally arises in many other fields of mathematics. The study of abstract residuated structures has originated from the investigation of ideal lattices of commutative rings with 1. In general, a partially ordered monoid is residuated if for all ab in its universe there exist $$a\rightarrow b = \max \{c: ca\le b\}$$ and $$a\rightsquigarrow b = \max \{c:ac\le b\}$$, and in other words, if for every a the translations $$x\rightarrow xa$$ and $$x\rightsquigarrow ax$$ are residuated mappings. If the multiplicative identity is the greatest element in the underlying order, then the monoid is integral. Residuation structures include lattice order groups and their negative cones as well as algebraic models of various propositional logics. In the logical context, the monoid operation $$\cdot$$ can be interpreted as conjunction and the residuals $$\rightarrow$$ and $$\rightsquigarrow$$ as two implications (they coincide if and only if the conjuncture is commutative).