Definition:Prime Element (Order Theory)
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Definition
Let $\struct {S, \wedge, \preceq}$ be a meet semilattice.
Let $p \in S$.
Then $p$ is a prime element (of $\struct {S, \wedge, \preceq}$) if and only if:
- $\forall x, y \in S: \paren {x \wedge y \preceq p \implies x \preceq p \text { or } y \preceq p}$
Also known as
A prime element of $\struct {S, \wedge, \preceq}$ can also be described as prime in $\struct {S, \wedge, \preceq}$.
Sources
- 1980: G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M.W. Mislove and D.S. Scott: A Compendium of Continuous Lattices
- Mizar article WAYBEL_6:def 6