Partially ordered set: Difference between revisions

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==Intervals==

An '''interval''' orA '''convex set''' in a poset ''P'' is a subset {{mvar|I}} of ''P'' with the property that, for any ''x'' and ''y'' in {{mvar|I}} and any ''z'' in ''P'', if ''x'' ≤ ''z'' ≤ ''y'', then ''z'' is also in {{mvar|I}}. This definition generalizes the definition of [[interval (mathematics)|interval]]s of [[real number]]s. When there is possible confusion with [[convex set]]s of [[geometry]], one uses '''order-convex''' instead of "convex".

A '''convex sulattice''' of a [[lattice (order theory)|lattice]] ''L'' is a sublattice of a lattice that is also a convex set of ''L''. Every nonempty convex sublattice can be uniquely represented as the intersection of a [[filter (mathematics)|filter]] and an [[ideal (order theory)|ideal]] of the lattice.

Authors who use the term "convex set" restrict often the term "interval" to the convex sets that can be defined with interval notation (as follows).

An '''interval''' in a poset ''P'' is a subset that can be defined with interval notation (as follows). Every interval is a convex set, but the converse does not hold.

For ''a'' ≤ ''b'', the ''closed interval'' {{closed-closed|''a'', ''b''}} is the set of elements ''x'' satisfying ''a'' ≤ ''x'' ≤ ''b'' (that is, ''a'' ≤ ''x'' and ''x'' ≤ ''b''). It contains at least the elements ''a'' and ''b''.