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单词 ASemilatticeIsACommutativeBand
释义

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.

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更新时间:2025/5/4 5:49:29