partially ordered algebraic system
Let be a poset. Recall a function on is said to be
order-preserving (or isotone) provided that , or
order-reversing (or antitone) provided that , or
whenever . Furthermore, is called monotone if is either isotone or antitone.
For every function on , 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),
The notion above can be generalized to -ary operations on a poset . An -ary operation on a poset is said to be isotone, antitone, or monotone iff when is isotone, antitone, or monotone with respect to each of its variables. We continue to use to arrow notations above to denote -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 such that is a poset, and every operation on is monotone. A partially ordered algebraic system is also called a partially ordered algebra, or a po-algebra for short.
Examples of po-algebras are po-groups, po-rings, and po-semigroups. In all three cases, the multiplication 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 over a field . The underlying universe is (not ). Addition over is, like the other examples above, isotone. Each element acts as a unary operator on , given by , the scalar multiplication of and . As is itself a poset, it can be partitioned into three sets: the positive cone of , the negative cone , and . Then iff it is as a unary operator, iff it is , and iff it is .
A homomorphism from one po-algebra to another is an isotone map from posets to that is at the same time a homomorphism from the algebraic systems to .
- 1 L. Fuchs, Partially Ordered Algebraic Systems, Addison-Wesley, (1963).
|Title||partially ordered algebraic system|
|Date of creation||2013-03-22 19:03:19|
|Last modified on||2013-03-22 19:03:19|
|Last modified by||CWoo (3771)|