# Category:Prime Elements

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This category contains results about Prime Elements in the context of Order Theory.

Let $\left({S, \wedge, \preceq}\right)$ be a meet semilattice.

Let $p \in S$.

Then $p$ is **prime (element)** if and only if

- $\forall x, y \in S: \left({ x \wedge y \preceq p \implies x \preceq p \lor y \preceq p }\right)$

## Pages in category "Prime Elements"

The following 19 pages are in this category, out of 19 total.

### C

- Characterization of Prime Element in Inclusion Ordered Set of Topology
- Characterization of Prime Element in Meet Semilattice
- Characterization of Pseudoprime Element by Finite Infima
- Characterization of Pseudoprime Element when Way Below Relation is Multiplicative
- Complement of Irreducible Topological Subset is Prime Element
- Complement of Lower Closure is Prime Element in Inclusion Ordered Set of Scott Sigma

### P

- Prime Element iff Complement of Lower Closure is Filter
- Prime Element iff Element Greater is Top
- Prime Element iff Meet Irreducible in Distributive Lattice
- Prime Element iff There Exists Way Below Open Filter which Complement has Maximum
- Prime Element is Meet Irreducible
- Prime Ideal is Prime Element
- Prime is Pseudoprime (Order Theory)
- Pseudoprime Element is Prime in Arithmetic Lattice