In the following, partial orders will usually just be denoted by their carrier sets. As long as the intended meaning is clear from the context, ≤ will suffice to denote the corresponding relational symbol, even without prior introduction. Furthermore, < will denote the strict order induced by ≤.

• Adjoint. See Galois connection.

• Algebraic poset. A poset is algebraic if it has a base of compact elements.

• Antichain. An antichain is a poset in which no two elements are comparable, i.e., there are no two distinct elements x and y such that x ≤ y. In other words, the order relation of an antichain is just the identity relation.

• Approximates relation. See way-below relation.

• A relation R on a set X is antisymmetric, if x R y and y R x implies x = y, for all elements x, y in X.

• An antitone function f between posets P and Q is a function for which, for all elements x, y of P, x ≤ y (in P) implies f(y) ≤ f(x) (in Q). Other names for this property is order-reversing and decreasing.

• An asymmetric relation R is a relation that is not symmetric.

• An atom in a poset P with least element 0, is an element that is minimal among all elements that are unequal to 0.

• A atomic poset P with least element 0 is one in which, for every non-zero element x of P, there is an atom a of P with a ≤ x.

• Base. See continuous poset.

• A Boolean algebra is a distributive lattice with least element 0 and greatest element 1, in which every element x has a complement ¬x, such that x ∧ ¬x = 0 and x ∨ ¬x = 1.

• A bounded poset is one that has a least element 0 and a greatest element 1.

• Chain. A chain is a totally ordered set or a totally ordered subset of a poset. See also total order.

• Closure operator. A closure operator on the poset P is a function C : P → P that is monotone, idempotent, and satisfies C(x) ≥ x for all x in P.

• Compact. An element x of a poset is compact if it is way below itself, i.e. x<<x. One also says that such an x is finite.

• Comparable. Two elements x and y of a poset P are comparable if either x ≤ y or y ≤ x.

• A complete Boolean algebra is a Boolean algebra that is a complete lattice.

• Complete Heyting algebra. A Heyting algebra that is a complete lattice is called a complete Heyting algebra. This notion coincides with the concepts frame and locale.

• Complete lattice. A complete lattice is a poset in which arbitrary (possibly infinite) joins (suprema) and meets (infima) exist.

• Complete partial order. A complete partial order, or cpo, is a directed complete poset with least element.

• Complete semilattice. A complete meet-semilattice is a poset in which arbitrary (possibly infinite) non-empty meets (infima) and all directed joins (suprema) exist. Complete join-semilattices are defined dually.

• Continuous poset. A poset is continuous if it has a base, i.e. a subset B of P such that every element x of P is the supremum of a directed set contained in {y in B | y<<x}.

• Continuous function. See Scott-continuous.

• dcpo. See directed complete.

• A dense poset P is one in which, for all elements x and y in P with x < y, there is an element z in P, such that x < z < y. A subset Q of P is dense in P if for any elements x < y in P, there is an element z in Q such that x < z < y.

• Directed. A non-empty subset X of a poset P is called directed, if, for all elements x and y of X, there is an element z of X such that x ≤ z and y ≤ z. The dual notion is called filtered.

• Directed complete. A poset D is said to be a directed complete poset, or dcpo, if every directed subset of D has a supremum.

• Distributive. A lattice L is called distributive if, for all x, y, and z in L, we find that x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z). This condition is known to be equivalent to its order dual.

• Domain. Domain is a general term for objects like those that are studied in domain theory. If used, it requires further definition.

• Down-set. See lower set.

• Dual. For a poset (P, ≤), the dual order (P, ≥) is defined by setting x ≥ y iff y ≤ x. The dual order of P is sometimes denoted by P^op, and is also called opposite or converse order. Any order theoretic notion induces a dual notion, defined by applying the original statement to the order dual of a given set. This exchanges ≤ and ≥, meets and joins, zero and unit.

• Filter. A subset X of a poset P is called a filter if it is a filtered upper set. The dual notion is called ideal.

• Filtered. A non-empty subset X of a poset P is called filtered, if, for all elements x and y of X, there is an element z of X such that z ≤ x and z ≤ y. The dual notion is called directed.

• Finite element. See compact.

• Frame. A frame F is a complete lattice, in which, for every x in F and every subset Y of F, the infinite distributive law x ∧ VY = V{x ∧ y | y in Y} holds. Frames are also known as locales and as complete Heyting algebras.

• Galois connection. Given two posets P and Q, a pair of monotone functions F:P → Q and G:Q → P is called a Galois connection, if F(x) ≤ y is equivalent to x ≤ G(y), for all x in P and y in Q. F is called the lower adjoint of G and G is called the upper adjoint of F.

• Greatest element. For a subset X of a poset P, an element a of X is called the greatest element of X, if x ≤ a for every element x in X. The dual notion is called least element.

• Heyting algebra. A Heyting algebra H is a bounded lattice in which the function f_a: H → H, given by f_a(x) = a ∧ x is the lower adjoint of a Galois connection, for every element a of H. The upper adjoint of f_a is then denoted by g_a, with g_a(x) = a ⇒ x. Every Boolean algebra is a Heyting algebra.

• An ideal is a subset X of a poset P that is a directed lower set. The dual notion is called filter.

• The incidence algebra of a poset is the associative algebra of all scalar-valued functions on intervals, with addition and scalar multiplication defined pointwise, and multiplication defined as a certain convoluation; see incidence algebra for the details.

• Infimum. For a poset P and a subset X of P, the greatest element in the set of lower bounds of X (if it exists, which it may not) is called the infimum, meet, or greatest lower bound of X. It is denoted by inf X or ^X. The infimum of two elements may be written as inf{x,y} or x ∧ y. If the set X is finite, one speaks of a finite infimum. The dual notion is called supremum.

• Irreflexive. A relation R on a set X is irreflexive, if there is no element x in X such that x R x.

• Join. See supremum.

• Lattice. A lattice is a poset in which all non-empty finite joins (suprema) and meets (infima) exist.

• Least element. For a subset X of a poset P, an element a of X is called the least element of X, if a ≤ x for every element x in X. The dual notion is called greatest element.

• Linear. See total order.

• Lower bound. A lower bound of a subset X of a poset P is an element b of P, such that b ≤ x, for all x in X. The dual notion is called upper bound.

• Lower set. A subset X of a poset P is called a lower set if, for all elements x in X and p in P, p ≤ x implies that p is contained in X. The dual notion is called upper set.

• Maximal element. A maximal element of a subset X of a poset P is an element m of X, such that m ≤ x implies m = x, for all x in X. The dual notion is called minimal element.

• Meet. See infimum.

• Minimal element. A minimal element of a subset X of a poset P is an element m of X, such that x ≤ m implies m = x, for all x in X. The dual notion is called maximal element.

• Monotone. A function f between posets P and Q is called monotone if, for all elements x, y of P, x ≤ y (in P) implies f(x) ≤ f(y) (in Q). Another name for this property is order-preserving.

• Order-isomorphism. A mapping f: P &rarr Q between two posets P and Q is called an order-isomorphism, if it is bijective and both f and f^-1 are monotone.

• Order-preserving. See monotone.

• Order-reversing. See antitone.

• Partial order. A partial order is a relation that is reflexive, antisymmetric, and transitive. Since both notions depend on each other, the term is also used to refer to a partially ordered set.

• Partially ordered set. A partially ordered set, or poset for short, is a set P together with a partial order ≤ defined on P.

• poset. See partial order.

• Preorder. A preorder is a relation that is reflexive and transitive. Such orders may also be called quasiorders.

• Preserving. A function f between posets P and Q is said to preserve suprema (joins), if, for all subsets X of P that have a supremum sup X in P, we find that sup{f(x): x in X} exists and is equal to f(sup X). Such a function is also called join-preserving. Analogously, one says that f preserves finite, non-empty, directed, or arbitrary joins (or meets). The converse property is called join-reflecting.

• Prime. An ideal I in a lattice L is said to be prime, if, for all elements x and y in L, x ∧ y in I implies x in I or y in I. The dual notion is called a prime filter. Equivalently, a set is a prime filter iff its complement is a prime ideal.

• Principal. A filter is called principal filter if it has a least element. Dually, a principal ideal is an ideal with a greatest element. The least or greatest elements may also be called principal elements in these situations.

• Pseudo-complement. In a Heyting algebra, the element x ⇒ 0 is called the pseudo-complement of x. It is also given by sup{y : y ∧ x = 0}, i.e. as the least upper bound of all elements y with y ∧ x = 0.

• Quasiorder. See preorder.

• Reflecting. A function f between posets P and Q is said to reflect suprema (joins), if, for all subsets X of P for which the supremum sup{f(x): x in X} exists and is of the form 'f(s) for some s in P, then we find that sup X exists and that sup X = s . Analogously, one says that f reflects finite, non-empty, directed, or arbitrary joins (or meets). The converse property is called join-preserving''.

• Reflexive. A relation R on a set X is reflexive, if x R x holds for all elements x, y in X.

• Scott-continuous. A monotone function f : P → Q between posets P and Q is Scott-continuous if, for every directed set D that has a supremum sup D in P, the set {fx | x in D} has the supremum f(sup D) in Q.

• Scott open. See Scott-topology.

• Scott topology. For a poset P, an upper set O is Scott-open if all directed sets D that have a supremum in O have non-empty intersection with O. The set of all Scott-open sets forms a topology, the Scott-topology.

• Semilattice. A semilattice is a poset in which either all finite non-empty joins (suprema) or all finite non-empty meets (infima) exist. Accordingly, one speaks of a join-semilattice or meet-semilattice.

• Smallest element. See least element.

• Strict order. A strict order is a relation that is antisymmetric, transitive, and irreflexive.

• Supremum. For a poset P and a subset X of P, the least element in the set of upper bounds of X (if it exists, which it may not) is called the supremum, join, or least upper bound of X. It is denoted by sup X or VX. The supremum of two elements may be written as sup{x,y} or x ∨ y. If the set X is finite, one speaks of a finite supremum. The dual notion is called infimum.

• Symmetric. A relation R on a set X is symmetric, if x R y implies y R x, for all elements x, y in X.

• Top. See unit.

• Total order. A total order T is a partial order in which, for each x and y in T, we have x ≤ y or y ≤ x. Total orders are also called linear orders or chains.

• Transitive. A relation R on a set X is transitive, if x R y and y R z imply x R z, for all elements x, y, z in X.

• Unit. The greatest element of a poset P can be called unit or just 1 (if it exists). Another common term for this element is top. It is the infimum of the empty set and the supremum of P. The dual notion is called zero.

• Up-set. See upper set.

• Upper bound. An upper bound of a subset X of a poset P is an element b of P, such that x ≤ b, for all x in X. The dual notion is called lower bound.

• Upper set. A subset X of a poset P is called an upper set if, for all elements x in X and p in P, p ≤ x implies that p is contained in X. The dual notion is called lower set.

• Way-below relation. In a poset P, some element x is way below y, written a<<b, if for all directed subsets D of P which have a supremum, y ≤ x implies y ≤ d for some d in D. One also says that x approximates y. See also domain theory.

• Zero. The least element of a poset P can be called zero or just 0 (if it exists). Another common term for this element is bottom. Zero is the supremum of the empty set and the infimum of P. The dual notion is called unit.