Conical residuated lattice-ordered idempotent monoids
A residuated lattice (r.l.) is an algebra \((L,\wedge,\vee,\cdot, e,\triangleright,\triangleleft)\) such that (i) \((L,\wedge,\vee)\) is a lattice, (ii) \((L,\cdot)\) is a semigroup with identity \(e\), (iii) the induced partial order on \(L\) is two-sided compatible with ``\(.\)'', and (iv) for all \(a,b\in L\), both \(\{x\in L\mid a\cdot x\leq b\}\) and \(\{y\in L\mid y\cdot a\leq b\}\) contain a greatest element denoted by \(a\triangleright b\) and \(b\triangleleft a\), respectively: An r.l. \(L\) is called conical if \((L,\cdot)\) is a band and for all \(a\in L\), either \(a\leq e\) or \(a\geq e\) (in this paper such an algebraic structure is also called a conical lattice-ordered idempotent monoid). Several properties of conical r.l.s are proved. As the main result a (quite involved) construction of all conical residuated lattices is given. Also the subdirectly irreducible conical r.l.s are characterized; the case that \((L,\cdot)\) is a semilattice (i.e., a commutative band) is dealt with, too. Finally, a characterization of conical r.l.s \(L\) which are simple (i.e., \(L\) does not contain a proper convex normal subalgebra) is provided. Also, the particular case of strictly simple r.l.s \(L\) is considered (i.e., \(L\) lacks nontrivial proper subalgebras and congruences).
scientific article; zbMATH DE number 5614946
scientific article