IRREDUCIBLE IDEALS IN BCI-ALGEBRAS
Yisheng Huang · Demonstratio Mathematica · 2004
In this paper, we show some results of irreducible ideals in BCI-algebras, including that every proper ideal of a BCK-algebra can be decomposed as the intersection of all minimal irreducible ideals associated with it, etc.Palasiriski showed that for lower BCK-semilattices, the notion of prime ideals is the same as that of irreducible ideals (see [5]) and he also investigated the decomposition of ideals as the intersection of prime ideals in lower BCK-semilattices (see [6]).In this paper, based on Palasinski's ideas, we will give some properties of irreducible ideals in BCI-algebras, and then consider the decomposition of ideals as the intersection of irreducible ideals in BCK-algebras.We will see that the main results in this paper are similar to the corresponding results in [5] and [6].Let us recall some definitions and results.For the notion of BCI-algebras, we refer the reader to [2].Assume that X is a BCI-algebra.It is known from [2] that the following identity holds:and that (X; <) is a partially ordered set where < is called the BCI-ordering on X whose definition is as follows: x < y if and only if x * y = 0 for any x, y € X.The algebra X is called a BCK-algebra if 0 * x = 0 for all x e X.If X is not a BCK-algebra, we call it a proper BCI-algebra.A lower BCK-semilattice X means that it is a BCK-algebra with (X; <) as a lower semilattice.