Definition:Meet (Order Theory)
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This page is about Meet in the context of Order Theory. For other uses, see Meet.
Definition
Let $\struct {S, \preceq}$ be an ordered set.
Let $a, b \in S$, and suppose that their infimum $\inf \set {a, b}$ exists in $S$.
Then $a \wedge b$, the meet of $a$ and $b$, is defined as:
- $a \wedge b = \inf \set {a, b}$
Expanding the definition of infimum, one sees that $c = a \wedge b$ if and only if:
- $(1): \quad c \preceq a$ and $c \preceq b$
- $(2): \quad \forall s \in S: s \preceq a$ and $s \preceq b \implies s \preceq c$
Also known as
Some sources refer to this as the intersection of $a$ and $b$.
Also see
- Results about the meet operation can be found here.
Sources
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 7$