On Komori Algebras

Yuichi KOMORI

                       komori@math.s.chiba-u.ac.jp

Department of Mathematics, Faculty of Science, Chiba University Inage-ku Chiba 263-8522 JAPAN

Humberstone([2]) has defined the notion of Komori algebras and proved a generalization of Lemma 2 of [3]. In this note, we will define Komori Algebras as a broader class of algebras than Komori algebras by Humberstone. And we will make the result of Humberstone more general.

We begin with the definition of Komori algebras. A Komori algebra is an algebra ⟨A, →, 1⟩ of type ⟨2, 0⟩ in which a quasi-identity (Ko) are satisfied:

(Ko) if x → y = y → x = 1, then x = y.

Henceforth Komori algebras defined by Humberstone are called Humberstone algebras. A Humberstone algebras is a Komori algebras which is satisfying the following three identities (M),(I) and (K):

\begin{aligned}(M) & 1 → x & = & x \\ (I) & x → x & = & 1 \\ (K) & x → 1 & = & 1. \end{aligned}

A BIK^{+}-algebra is a Humberstone algebra which is satisfying the following identity (B):

(B) (y → z) → (x → y) → x → z = 1.

A BIK^{+}-algebra had been called a BCC-algebra in [3]. In that paper, we proved that the quasivariety of all BIK^{+}-algebras is not a variety. A BCK-algebra ia a BIK^{+}-algebra which is satisfying the following identity (C):

(C) x → (y → z) = y → (x → z).

Wronski([5]) proved that the quasivarieties of all BCK-algebras is not a variety.

Now we define two finite algebras NK and K4. The algebra NK is closely related to the partition of the set of terms constructed out of the variables x, y (and the constant 1) defined in [3]. The algebras NK and K4 are defined by Hasse diagrams Figure 1 and Figure 2 with the definition of a binary operation →, respectively. The algebra NK is not a Komori algebra. The algebra K4 is a BIK^{+}-algebra, but not a BCK-algebra because c → (a → b) = 1 ≠ a → (c → b) = c. Let f be a function from K4 onto NK such that f(1) = f(c) = 1, f(a) = a and f(b) = b. Then f is a homomorphism from K4 onto NK. So we have the following.

Theorem 1 If a class K of Komori algebras contains K4, then K is not a variety.

Corollary 2 The class of all BIK^{+}-algebras is not a variety.

[figure: a; 1; b] \{ \begin{matrix}1 → x = x, \\ x → y = 1 if x ≤ y, \\ a → b = b → a = 1. \end{matrix}

Figure 1: NK

[figure: a; c; 1; b] \{ \begin{matrix}1 → x = x, \\ x → y = 1 if x ≤ y, \\ a → b = b → a = c, \\ c → a = a, c → b = b. \end{matrix}

Figure 2: K4

Let WR be the algebra defined by Wronski to prove that the class of all BCK-algebras is not a variety.

Theorem 3 If a class K of Komori algebras contains WR, then K is not a variety.

Corollary 4 The class of all BCK-algebras is not a variety.

The above corollaries have been proved in [3] and [5], respectively. Corollary 2 follows from Corollary 4 becase a class K of Komori algebras is not a variety if Id(K) ⊆ Σ and {A|Σ ⊆ Id(A) and A is a Komori algebra} is not a variety, in general. Here Id(K) is the set of identities holding in all algebras in K and Id(A) = Id({A}). Theorem 1 and Theorem 3 are mutually independent becasue x → (x → y) = x → (x → (x → y)) ∈ Id(K4) and the identity ∉ Id(WR). We call such algebras in the above theorems criterions.

Definition 5 A Komori algebra A is a criterion if any class of Komori algebras containing A is not variety.

For any Komori algebras A and B, if Id(A) ⊆ Id(B) and B is a criterion, then A is a criterion. For any criterion A, by Zorn’s lemma, there exists a criterion B such that Id(A) ⊆ Id(B) and C is not a criterion if Id(B) ⊊ Id(C). We call such criterion B maximal.

Conjecture 6 The Komori algebras K4 and WR are both maximal, that is, any Komori algebra A is not a criterion if Id(K4) ⊊ Id(A) or Id(WR) ⊊ Id(A).

Now we describe a partition in [3] of the set of terms constructed out of the variables x, y (and the constant 1), whose blocks are called X, Y, and 1. The partition in question is defined by induction on the construction of these x, y-terms:

1. x ∈ X, y ∈ Y, and 1 ∈ 1,

2. if s ∈ X or s ∈ Y, then s → t ∈ 1,

3. if s ∈ 1, then s → t belongs to the block which t belongs to.

Let XY be the set of identities {s = t|s ∈ X and t ∈ Y}.

Lemma 7 Let K be a class of Komori algebras. If x, y-terms s_{1} and s_{2} lie in the different block of the partition {X, Y, 1} and s_{1} = s_{2} ∈ Id(K), then

Id(K) ∩ XY ≠ ∅ .

Let K be a class of Komori algebras and F_{K}(2) be the K-free algebra on two generators(x and y). Suppose that Id(K) ∩ XY = ∅. Let f be a function from F_{K}(2) onto NK defined below:

f([t]) = \{ \begin{matrix}1 & if t ∈ 1, \\ a & if t ∈ X, \\ b & if t ∈ Y. \end{matrix}

By Lemma 7, the function f is well-defined. Remark that Lemma 7 and Id(K)∩XY = ∅ are used only for establishing the well-definedness of the function f. The function f is a homomorphism from F_{K}(2) onto NK. So the algebra F_{K}(2) is a criterion.

Theorem 8 Let K be a class of Komori algebras. If Id(K) ∩ XY = ∅, then the class K is not a variety.

The above theorem is a generalization of Lemma 2 of [3] and Proposition 3 of [2]. Remark that a identity in the set XY is not always a Komori identity (in the sence of [1]) for the quasi-variety of all Komori algebras though the set XY is a complete set of Komori identies for the quasi-variety of all Humberstone algebras.

References

[1] W. J. Blok and Silvia B. La Falce. Komori identities in algebraic logic. to appear in Reports on Mathematical Logic, 2000.

[2] Lloyd Humberstone. Note on a lemma of komori. submitted to Mathematica Japonica, 2000.

[3] Yuichi Komori. The class of BCC–algebras is not a variety. Mathematica Japonica, 29:391– 394, 1984.

[4] Misao Nagayama. On a property of BCK-identities. Studia Logica, 53:227–234, 1994.

[5] Andrzej Wroński. BCK-algebras do not form a variety. Mathematica Japonica, 28:211–213, 1983.