a semilattice is a commutative band

This note explains how a semilattice is the same as a commutativePlanetmathPlanetmathPlanetmath band.

Let S be a semilattice, with partial orderMathworldPlanetmath < and each pair of elements x and y having a greatest lower boundMathworldPlanetmath xy. Then it is easy to see that the operationMathworldPlanetmath defines a binary operationMathworldPlanetmath on S which makes it a commutative semigroup, and that every element is idempotentMathworldPlanetmathPlanetmath since xx=x.

Conversely, if S is such a semigroupPlanetmathPlanetmath, define xy iff x=xy. Again, it is easy to see that this defines a partial order on S, and that greatest lower bounds exist with respect to this partial order, and that in fact xy=xy.

Title a semilattice is a commutative band
Canonical name ASemilatticeIsACommutativeBand
Date of creation 2013-03-22 12:57:28
Last modified on 2013-03-22 12:57:28
Owner mclase (549)
Last modified by mclase (549)
Numerical id 6
Author mclase (549)
Entry type Proof
Classification msc 20M99
Classification msc 06A12
Related topic LatticeMathworldPlanetmath