BCI algebra

An algebra (in the sense of universal algebra) ${\displaystyle \left(X;\ast ,0\right)}$ of type ${\displaystyle \left(2,0\right)}$ is called a BCI-algebra if, for any ${\displaystyle x,y,z\in X}$, it satisfies the following conditions. (Informally, we may read ${\displaystyle 0}$ as "truth" and ${\displaystyle x\ast y}$ as "${\displaystyle y}$ implies ${\displaystyle x}$".)

BCI-1

${\displaystyle \left(\left(x\ast y\right)\ast \left(x\ast z\right)\right)\ast \left(z\ast y\right)=0}$

BCI-2

${\displaystyle \left(x\ast \left(x\ast y\right)\right)\ast y=0}$

BCI-3

${\displaystyle x\ast x=0}$

BCI-4

${\displaystyle x\ast y=0\land y\ast x=0\implies x=y}$

BCI-5

${\displaystyle x\ast 0=0\implies x=0}$

BCK algebra

A BCI-algebra ${\displaystyle \left(X;\ast ,0\right)}$ is called a BCK-algebra if it satisfies the following condition:

BCK-1

${\displaystyle \forall x\in X:0\ast x=0.}$

A partial order can then be defined as x ≤ y iff x * y = 0.

A BCK-algebra is said to be commutative if it satisfies:

${\displaystyle x\ast (x\ast y)=y\ast (y\ast x)}$

In a commutative BCK-algebra x * (x * y) = x ∧ y is the greatest lower bound of x and y under the partial order ≤.

A BCK-algebra is said to be bounded if it has a largest element, usually denoted by 1. In a bounded commutative BCK-algebra the least upper bound of two elements satisfies x ∨ y = 1 * ((1 * x) ∧ (1 * y)); that makes it a distributive lattice.