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join (Definition)

Certain posets $ X$ have a binary operation join denoted by $ \lor$, such that $ x \lor y$ is the least upper bound of $ x$ and $ y$. Such posets are called join-semilattices, or $ \lor$-semilattices, or upper semilattices.

If $ j$ and $ j'$ are both joins of $ x$ and $ y$, then $ j \leq j'$ and $ j' \leq j$, and so $ j = j'$; thus a join, if it exists, is unique. The join is also known as the or operator.



"join" is owned by yark. [ full author list (2) | owner history (1) ]
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See Also: meet, semilattice

Other names:  or operator
Also defines:  join-semilattice, join semilattice, upper semilattice
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Cross-references: least upper bound, binary operation, posets
There are 48 references to this entry.

This is version 8 of join, born on 2002-02-24, modified 2005-02-26.
Object id is 2611, canonical name is Join.
Accessed 7224 times total.

Classification:
AMS MSC: 06A12 (Order, lattices, ordered algebraic structures :: Ordered sets :: Semilattices)
Pending Errata and Addenda
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