### Table of Contents

## Pocrims

Abbreviation: **Pocrim**

### Definition

A \emph{pocrim} (short for \emph{partially ordered commutative residuated integral monoid}) is a structure $\mathbf{A}=\langle A,\oplus,\ominus,0\rangle$ of type $\langle 2,2,0\rangle$ such that

(1): $((x \ominus y) \ominus (x \ominus z)) \ominus (z \ominus y) = 0$

(2): $x \ominus 0 = x$

(3): $0 \ominus x = 0$

(4): $(x \ominus y) \ominus z = x \ominus (z \oplus y)$

(5): $x \ominus y = y \ominus x = 0 \Longrightarrow x=y$

##### Morphisms

Let $\mathbf{A}$ and $\mathbf{B}$ be pocrims. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: $h(x \oplus y)=h(x) \oplus h(y)$, $h(x \ominus y)=h(x) \ominus h(y)$, $h(0)=0$.

### Definition

A \emph{pocrim} is a structure $\mathbf{A}=\langle A,\oplus,\ominus,0\rangle$ such that

$\langle A,\ominus,0\rangle$ is a BCK-algebra

$(x \ominus y) \ominus z = x \ominus (z \oplus y)$

### Examples

Example 1:

### Basic results

### Properties

Feel free to add or delete properties from this list. The list below may contain properties that are not relevant to the class that is being described.

### Finite members

1,1,2,7,26,129

$\begin{array}{lr}

f(1)= &1\\ f(2)= &1\\ f(3)= &2\\ f(4)= &7\\ f(5)= &26\\

\end{array}$ $\begin{array}{lr}

f(6)= &129\\ f(7)= &\\ f(8)= &\\ f(9)= &\\ f(10)= &\\

\end{array}$

### Subclasses

### Superclasses

### References

^{1)}D. Higgs, \emph{Dually residuated commutative monoids with identity element as least element do not form an equational class}, Math. Japon., \textbf{29}, 1984, no. 1, 69–75 MRreview