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Ordered monoids with zero

Abbreviation: OMonZ

Definition

An ordered monoid with zero is of the form $\mathbf{A}=\langle A,\cdot,1,0,\le\rangle$ such that $\mathbf{A}=\langle A,\cdot,1,\le\rangle$ is an ordered monoid and

$0$ is a zero: $x\cdot 0 = 0$ and $0\cdot x = 0$

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It is not unusual to give several (equivalent) definitions. Ideally, one of the definitions would give an irredundant axiomatization that does not refer to other classes.

Morphisms

Let $\mathbf{A}$ and $\mathbf{B}$ be ordered monoids. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a orderpreserving homomorphism: $h(x \cdot y)=h(x) \cdot h(y)$, $h(1)=1$, $h(0)=0$, $x\le y\Longrightarrow h(x)\le h(y)$.

Definition

A is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle ...\rangle$ such that

$...$ is …: $axiom$

$...$ is …: $axiom$

Examples

Example 1:

Basic results

Properties

Finite members

$f(n)=$ number of members of size $n$.

$\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &3\\ f(4)= &15\\ f(5)= &84\\ f(6)= &575\\ f(7)= &4687\\ f(8)= &45223\\ f(9)= &\\ \end{array}$

Subclasses

[[Commutative ordered monoids]]

Superclasses

[[Ordered monoids]] reduced type
[[Ordered semigroups with zero]] reduced type
[[Representable residuated lattices]]

References