Residuation is a fundamental concept of ordered structures and the residuated lattices, obtained by adding a residuated monoid operation to lattices, have been applied in several branches of mathematics, including ℓ-groups, ideal lattices of rings and multiple-valued logics. Since the late 1930s, the commutative residuated lattices have been studied by Krull, Dilworth and Ward.
Non-commutative residuated lattices, sometimes called pseudo-residuated lattices, biresiduated lattices or generalized residuated lattices, are the algebraic counterparts of substructural logics, i.e. logics which lack at least one of the three structural rules, namely contraction, weakening and exchange. Particular cases of residuated lattices are the full Lambek algebras (FL-algebras), integral residuated lattices and bounded integral residuated lattices (FL w -algebras).
In this chapter we investigate the properties of a residuated lattice and the lattice of filters of a residuated lattice, we study the Boolean center of an FL w -algebra and we define and study the directly indecomposable FL w -algebras. We prove that any linearly ordered FL w -algebra is directly indecomposable and any subdirectly irreducible FL w -algebra is directly indecomposable. Finally, the FL w -algebras of fractions relative to a meet-closed system is introduced and investigated.
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