Join and meet

Let \(\langle P, \leq \rangle\) be a partially ordered set, and let \(S \subseteq P\). The join of \(S\) in \(P\), denoted by \(\bigvee_P S\), is an element \(p \in P\) satisfying the following two properties:

  • p is an upper bound of \(S\); that is, for all \(s \in S\), \(s \leq p\).

  • For all upper bounds \(q\) of \(S\) in \(P\), \(p \leq q\).

\(\bigvee_P S\) does not necessarily exist, but if it does then it is unique. The notation \(\bigvee S\) is typically used instead of \(\bigvee_P S\) when \(P\) is clear from context. Joins are often called least upper bounds or supremums. For \(a, b\) in \(P\), the join of \(\{a,b\}\) in \(P\) is denoted by \(a \vee_P b\), or \(a \vee b\) when \(P\) is clear from context.

The dual concept of the join is that of the meet. The meet of \(S\) in \(P\), denoted by \(\bigwedge_P S\), is defined an element \(p \in P\) satisfying.

  • p is a lower bound of \(S\); that is, for all \(s\) in \(S\), \(p \leq s\).

  • For all lower bounds \(q\) of \(S\) in \(P\), \(q \leq p\).

Meets are also called infimums, or greatest lower bounds. The notations \(\bigwedge S\), \(p \wedge_P q\), and \(p \wedge q\) are all have meanings that are completely analogous to the aforementioned notations for joins.

Basic example

The above Hasse diagram represents a poset with elements \(a\), \(b\), \(c\), and \(d\). \(\bigvee \{a,b\}\) does not exist because the set \(\{a,b\}\) has no upper bounds.\(\bigvee \{c,d\}\) does not exist for a different reason: although \(\{c, d\}\) has upper bounds \(a\) and \(b\), these upper bounds are incomparable, and so \(\{c, d\}\) has no least upper bound. There do exist subsets of this poset which possess joins; for example, \(a \vee c = a\), \(\bigvee \{b,c,d\} = b\), and \(\bigvee \{c\} = c\).

Now for some examples of meets.\(\bigwedge \{a, b, c, d\}\) does not exist because \(c\) and \(d\) have no common lower bounds. However, \(\bigwedge \{a,b,d\} = d\) and \(a \wedge c = c\).

Additional Material

Further reading

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  • Order theory

    The study of binary relations that are reflexive, transitive, and antisymmetic.