partially ordered algebraic system

Let A be a poset. Recall a function f on A is said to be

  • order-preserving (or isotone) provided that f(a)f(b), or

  • order-reversing (or antitone) provided that f(a)f(b), or

whenever ab. Furthermore, f is called monotone if f is either isotone or antitone.

For every function f on A, we denote it to be , , or according to whether it is isotone, antitone, or both. The following are some easy consequences:

  • == (meaning that the composition of an isotone and an antitone maps is antitone),

  • == (meaning that the composition of two isotone or two antitone maps is isotone),

  • f is iff it is a constant on any chain in A, and if this is the case, for every aA, f-1(a) is a maximal chain in A.

The notion above can be generalized to n-ary operationsMathworldPlanetmath on a poset A. An n-ary operation f on a poset A is said to be isotone, antitone, or monotone iff when f is isotone, antitone, or monotone with respect to each of its n variables. We continue to use to arrow notations above to denote n-ary monotone functions. For example, a ternary function that is (,,) is isotone with respect to its first and third variables, and antitone with respect to its second variable.

Definition. A partially ordered algebraic system is an algebraic system 𝒜=(A,O) such that A is a poset, and every operation fO on A is monotone. A partially ordered algebraic system is also called a partially ordered algebraMathworldPlanetmathPlanetmath, or a po-algebra for short.

Examples of po-algebras are po-groups, po-rings, and po-semigroups. In all three cases, the multiplicationPlanetmathPlanetmath operations are (,), as well as the addition operation in a po-ring.. In the case of a po-group, the multiplicative inverse operation is , as well as the additive inverse operation in a po-ring.

Another example is an ordered vector space V over a field k. The underlying universePlanetmathPlanetmath is V (not k). Addition over V is, like the other examples above, isotone. Each element rk acts as a unary operator on V, given by r(v)=rv, the scalar multiplication of r and v. As k is itself a poset, it can be partitioned into three sets: the positive cone P(k) of k, the negative cone -P(k), and {0}. Then rP(k) iff it is as a unary operator, r-P(k) iff it is , and r=0 iff it is .


  • A homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath from one po-algebra 𝒜 to another is an isotone map ϕ from posets A to B that is at the same time a homomorphism from the algebraic systems 𝒜 to .

  • A partially ordered subalgebraMathworldPlanetmathPlanetmath of a po-algebra 𝒜 is just a subalgebra of 𝒜 viewed as an algebra, where the partial ordering on the universe of the subalgebra is inherited from the partial ordering on A.


  • 1 L. Fuchs, Partially Ordered Algebraic Systems, Addison-Wesley, (1963).
Title partially ordered algebraic system
Canonical name PartiallyOrderedAlgebraicSystem
Date of creation 2013-03-22 19:03:19
Last modified on 2013-03-22 19:03:19
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 8
Author CWoo (3771)
Entry type Definition
Classification msc 06F99
Classification msc 08C99
Classification msc 08A99
Related topic AlgebraicSystem