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 $f(a)\le f(b)$, or

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

whenever $a\le b$. Furthermore, $f$ is called monotone if $f$ is either isotone or antitone.

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

• $\uparrow \circ \downarrow =\downarrow \circ \uparrow =\downarrow$ (meaning that the composition of an isotone and an antitone maps is antitone),

• $\uparrow \circ \uparrow =\downarrow \circ \downarrow =\uparrow$ (meaning that the composition of two isotone or two antitone maps is isotone),

• $f$ is $\updownarrow$ iff it is a constant on any chain in $A$, and if this is the case, for every $a\in A$, $f^{-1}(a)$ is a maximal chain in $A$.

The notion above can be generalized to $n$-ary operations 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 $(\uparrow ,\downarrow ,\uparrow )$ 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 $f\in O$ on $A$ 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 $(\uparrow ,\uparrow )$, as well as the addition operation in a po-ring.. In the case of a po-group, the multiplicative inverse operation is $\downarrow$, 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 universe is $V$ (not $k$). Addition over $V$ is, like the other examples above, isotone. Each element $r\in k$ 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 $r\in P(k)$ iff it is $\uparrow$ as a unary operator, $r\in -P(k)$ iff it is $\downarrow$, and $r=0$ iff it is $\updownarrow$.

Remarks

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

• A partially ordered subalgebra 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 $A$.

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