Order, lattices, ordered algebraic structures

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• ConceptOrder theory

  Order theory is the branch of mathematics that studies orders, that is, binary relations that compare elements, such as partial orders, total orders and preorders. Its basic objects are partially ordered sets and the monotone functions between them. It includes the theory of lattices, Galois connections and closure operators. Its ideas appear in algebra, topology, logic and computer science, wherever objects can be compared by size, inclusion or strength.

• ConceptBinary relation

  A binary relation between two sets says, for each pair of elements, whether the first element is related to the second. Formally, it is a set of ordered pairs.

• ConceptPreorder

  A preorder is a relation that is reflexive and transitive. It compares elements, but two different elements can be equivalent under it.

• ConceptPartial order

  A partial order is a relation that is reflexive, antisymmetric and transitive. A set with a partial order is a partially ordered set, or poset.

• ConceptStrict partial order

  A strict partial order is a binary relation $<$ that is irreflexive and transitive. It is the "strictly less than" version of a partial order. Strict partial orders on a set correspond one to one with partial orders on the same set. So either one can serve as the basic notion.

• ConceptTotal order

  A total order is a partial order in which any two elements are comparable.