partially ordered algebraic system
Let $A$ be a poset. Recall a function $f$ on $A$ is said to be

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orderpreserving (or isotone) provided that $f(a)\le f(b)$, or

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orderreversing (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:

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

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$\uparrow \circ \uparrow =\downarrow \circ \downarrow =\uparrow $ (meaning that the composition of two isotone or two antitone maps is isotone),
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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 $\mathcal{A}=(A,O)$ 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 poalgebra for short.
Examples of poalgebras are pogroups, porings, and posemigroups. In all three cases, the multiplication^{} operations are $(\uparrow ,\uparrow )$, as well as the addition operation in a poring.. In the case of a pogroup, the multiplicative inverse operation is $\downarrow $, as well as the additive inverse operation in a poring.
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

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A homomorphism^{} from one poalgebra $\mathcal{A}$ to another $\mathcal{B}$ is an isotone map $\varphi $ from posets $A$ to $B$ that is at the same time a homomorphism from the algebraic systems $\mathcal{A}$ to $\mathcal{B}$.

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A partially ordered subalgebra^{} of a poalgebra $\mathcal{A}$ is just a subalgebra of $\mathcal{A}$ viewed as an algebra, where the partial ordering on the universe of the subalgebra is inherited from the partial ordering on $A$.
References
 1 L. Fuchs, Partially Ordered Algebraic Systems, AddisonWesley, (1963).
Title  partially ordered algebraic system 

Canonical name  PartiallyOrderedAlgebraicSystem 
Date of creation  20130322 19:03:19 
Last modified on  20130322 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 