Pocrims

Abbreviation: Pocrim

Definition

A pocrim (short for 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 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

Finite members

$\begin{array}{lr} f(1)= &1\\ f(2)= &\\ f(3)= &\\ f(4)= &\\ f(5)= &\\ \end{array}$ $\begin{array}{lr} f(6)= &\\ f(7)= &\\ f(8)= &\\ f(9)= &\\ f(10)= &\\ \end{array}$

Subclasses

Superclasses

References


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