the amalgamation property for some classes of bck-algebras

Bulletin of the Section of Logic
Volume 14/3 (1985), pp. 109–112
reedition 2007 [original edition, pp. 109–113]
Katarzyna Pa̷lasińska
THE AMALGAMATION PROPERTY FOR SOME
CLASSES OF BCK-ALGEBRAS
This is an abstract of a paper which is to be submitted to Reports on
Mathematical Logic.
It is known, that the class of all BCK-algebras enjoys the strong
amalgamation property (see [6]). Here we present some results concerning
the amalgamation property (AP ) for some subclasses of the class of all
BCK-algebras.
The notations and the terminology used in this paper are rather standard.
For a background on universal algebra we refer the reader to G.
Grätzer [2] and for BCK-algebras to K. Iséki and S. Tanaka [4]. We let
N = {0, 1, . . .} and N + = N\{0}.
Let us recall that a BCK-algebra A is called commutative iff the
identity x ∗ (x ∗ y) = y ∗ (y ∗ x) holds in A. The class T of all commutative
BCK-algebras is a variety (see [7]). For n ∈ N + the class ξn is defined as
the class of all BCK-algebras satisfying the identity:
(En) x ∗ n y = x ∗ n+1 y,
where x ∗ 1 y = x ∗ y and x ∗ k+1 y = (x ∗ k y) ∗ y, for k ∈ N + . For n ∈ N +
the class ξn is a variety (see [1]).
For every n ∈ N + we define BCK-algebra ̷L n = < {0, 1, . . . , n − 1},
∗ ̷L , 0 > and H n = < {0, 1, . . . , n − 1}, ∗H, 0 > putting for every k, l =
0, . . . , n − 1, k ∗ ̷L l = max(0, k − l), k ∗H l = k if k > 1 and k ∗H l = 0
otherwise.
We have the following
Theorem 1. The variety ξ1 has (AP ).

110 Katarzyna Pa̷lasińska
This theorem can be proved by using some modification of Wroński’s
method in [6]. It is worth to mention that the variety” ξ1 ∩ T which
is determined by the 2-element BCK-chain ̷L 2 has (AP ) by virtue of a
theorem of Grätzer, Lakser [3] stating that of a variety V has the congruence
extension property and subalgebras of subdirectly irreducible algebras from
V are subdirectly irreducible then V has (AP ) iff every amalgam in the class
of subdirectly irreducible algebras from V can be amalgamated in V.
The next two theorems give necessary conditions for the class K of
BCK-algebras to have (AP ).
Theorem 2. If for some n ∈ N + , K ⊆ ξn, K is closed wit respect to the
operation of forming subalgebras and ̷L 3 ∈ K then K does not have (AP ).
Theorem 3. If K is closed with respect to the operations of forming
subalgebras and isomorphic copies, H 3 ∈ K and K has (AP ), then for
every n ∈ N + H n ∈ K.
get
Since ̷L 3 ∈ ξn ∩ T for n ∈ N + , n > 1, then by Theorems 1 and 2 we
Corollary 1. For any n ∈ N + , ξn has (AP ) iff n = 1.
Corollary 2. For any n ∈ N + , ξn ∩ T has (AP ) iff n = 1.
Another important consequence of Theorems 2 and 3 is the following
full characterization of those varieties generated by finite BCK-algebras
which have (AP ).
Theorem 4. For every variety V generated by a finite BCK-algebra, V
has (AP ) iff V is trivial or V = HSP (̷L 2).
To prove Theorem 4 let us note that the trivial variety of BCKalgebras
as well as HSP (L 2 ) = ξ1 ∩ T have (AP ). To prove the converse
implication we need the following:
Lemma 1. For any variety V of BCK-algebras generated by its finite
members, if V ⊆ HSP (̷L 2 ) then {H 3 , ̷L 3 } ∩ V = ∅.
Lemma 2. Every finite BCK-algebras satisfies an identity (En) for some
n ∈ N + .

The Amalgamation Property for Some Classes Of BCK-Algebras 111
Now, if V = HSP (A), where A is a finite BCK-algebra and ̷L 3 ∈ V
then Theorem 4 follows from Theorem 2. The case H 3 ∈ V needs some
more attention.
Following S. Nagata [5] for n ∈ N we define terms αn putting
α0 = x0
αn+1 = xn+1 ∗ [xn+1 ∗ (αn ∗ xn)]
and we have
Lemma 3. Every finite BCK-algebra satisfies an identity αn = 0 for some
n ∈ N.
Lemma 4. For m, n ∈ N, if m > n + 1 then αm = 0 fails to hold in H m .
Returning to the proof of Theorem 4 assume that V = HSP (A) where
A is a finite BCK-algebra and H 3 ∈ V. Let us suppose that V has (AP ).
Then by virtue of Theorem 3 H n ∈ V for every n ∈ N + . On the other
hand for some n ∈ N, αn = 0 is an identity of V and – as αn = 0 is not
satisfied in Hn+2 − Hn+2 ∈ V, a contradiction. Thus if H 3 ∈ V, then V
does not have (AP ).
References
[1] W. H. Cornish, Varieties generated by finite BCK-algebras, Bulletin
Australian Mathematical Society 22 (1980), pp. 411–430.
[2] G. Grätzer, Universal Algebra, Springer-Verlag, New York
(1979), 2nd edition.
[3] G. Grätzer and H. Lakser, The structure of pseudo complemented
distributive lattices, II, Congruence extension and amalgamation, Transactions
of the American Mathematical Society 156 (1971), pp. 343–
358.
[4] K. Iséki, S. Tanaka, An introduction to the theory of BCKalgebras,
Mathematica Japonica 23 (1978), pp. 1–26.
[5] S. Nagata, A series of succesive modification of Pierce’s rule,
Proceedings Japan Academy 42 (1966), pp. 859–861.
[6] A. Wroński, Interpretation and amalgamation properties of BCKalgebras,
Mathematica Japonica 29 (1984), pp. 115-121.

112 Katarzyna Pa̷lasińska
[7] H. Yutani, The class of commutative BCK-algebras is equationally
definable, Mathematics Seminar Notes 5 (1977), pp. 207-210.
Department of Logic
Jagiellonian University
Cracow, Poland