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[parent] a semilattice is a commutative band (Proof)

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

Let $ S$ be a semilattice, with partial order $ <$ and each pair of elements $ x$ and $ y$ having a greatest lower bound $ x \wedge y$. Then it is easy to see that the operation $ \wedge$ defines a binary operation on $ S$ which makes it a commutative semigroup, and that every element is idempotent since $ x \wedge x = x$.

Conversely, if $ S$ is such a semigroup, define $ x \leq y$ 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 $ x \wedge y = xy$.



"a semilattice is a commutative band" is owned by mclase.
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See Also: lattice


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Cross-references: iff, semigroup, idempotent, commutative semigroup, binary operation, operation, easy to see, greatest lower bound, partial order, band, commutative, semilattice
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This is version 3 of a semilattice is a commutative band, born on 2002-08-19, modified 2002-08-20.
Object id is 3320, canonical name is ASemilatticeIsACommutativeBand.
Accessed 1875 times total.

Classification:
AMS MSC: 20M99 (Group theory and generalizations :: Semigroups :: Miscellaneous)
 06A12 (Order, lattices, ordered algebraic structures :: Ordered sets :: Semilattices)
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