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

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 $\mathcal{A}=(A,O)$ 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 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 $\lbrace 0\rbrace$ . 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 $\mathcal{A}$ to another $\mathcal{B}$ is an isotone map $\phi$ from posets $A$ to $B$ that is at the same time a homomorphism from the algebraic systems $\mathcal{A}$ to $\mathcal{B}$ .
• A partially ordered subalgebra of a po-algebra $\mathcal{A}$ 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 $A$ .

Bibliography

1

L. Fuchs, Partially Ordered Algebraic Systems, Addison-Wesley, (1963).