🚀 go-pugleaf

Hi All,

Recall that a _BCK-algebra_ is a groupoid <A; *, 0> satisfying:

((x * y) * (x * z)) * (z * y) = 0       (1)
(x * (x * y)) * y = 0                   (2)
x * x = 0                               (3)
0 * x = 0                               (4)
x <= y, y <= x implies x = y            (5)
x <= y if and only if x * y = 0.        (6)

and that such an algebra also satisfies:

(x * y) * z = (x * z) * y.              (7)

Recall too that a BCK-algebra is called _positive implicative_ if
the equality:

(x * z) * (y * z) = (x * y) * z         (8)

holds, _commutative_ if the equality:

x * (x * y) = y * (y * x)               (9)

holds, and _implicative_ if the equality:

x * (y * x) = x                        (10)

holds.

Cornish has previously considered the class of _pre-BCK-algebras_.
These are groupoids <A; *, 0> such that (1)--(4) holds, but (5) fails.
Clearly pre-BCK-algebras form a variety.

Example 1.   Let <A; \, 0> be an algebra with a zero 0 and a binary
operation \ defined by:

a \ b = a if b = 0
        0 otherwise

for any a, b in A.   Then <A; \, 0> satisfies (1), (2), (3) and (4),
but not (5).   Thus <A; \, 0> is a pre-BCK-algebra.

Consider now the variety V of type <2, 0> defined by the following
equations:

(x * y) * z = (x * z) * y               (7)
(x * z) * (y * z) = (x * y) * z         (8)
x * (y * x) = x                        (10)
x * x = 0.                             (11)

It can be shown that V satisfies (1), (2), (3) and (4), but not (5).
Thus V is a class of pre-BCK-algebras.   The following BCK-algebra
identities also hold in V:

(x * y) * x = 0                        (12)
x * 0 = x                              (13)
0 * x = 0                              (14)
x * y = (x * y) * y                    (15)
x * (x * (x * y)) = x * y.             (16)

It is not difficult to show that (9) fails to hold in V; if (9) holds,
then V collapses to the variety of implicative BCK-algebras.

Some questions.   Has this class of algebras been investigated before?
If so, what are they called, and what is a pointer to the literature?

Comments?

Thanks,

- Matthew.

--
Matthew Spinks      |
PhD grad. slave     | 'I bet the human brain is a kludge.'
GSCIT Research Ctr. |
Monash University   | - Marvin Minsky
Churchill, Victoria |
Australia           |