ON THE UNIQUE AXIOM OF BCII CLASS

Jacek K. Kabzi´ nski

ON THE UNIQUE AXIOM OF BCII CLASS

In the paper [2] we defined BCII variety constructing a subclass of BCI quasi variety determined by K. Iseki in [1]. BCII variety polynomially eqivalent to the class of Abelian groups is a natural semantics for BCI consequence of identity connective (see [4],[3]). The aim of this paper is to present the unique axiom determining BCII variety and due to the definability of the constant 0 in the algebras of this variety, its polynomially equivalent counterpart in the class of algebras of type < 2 >. In the paper we apply the convention of associating to the left and ignoring the symbol of this binary operation. Let us recall that in [2] the variety of BCII algebras was determined by the following identities: (∗1) (∗2)

a0 = a, ab(ac) = cb. Let as consider the identity:

(1)

ab(ac)(cxb) = x.

From the above identity we infer identities (∗1) and (∗2). The converse deduction is straightforward, however it is still easier to show that the result of transformation of equality (1) (cf. [2]) is an identity of Abelian groups variety. Subsequently we infer: (1)

ab(ac)(cxb) = x

(2)

= (1)[a/d, b/cxb, c/ab, x/ac] d(cxb)(d(ab))(ab(ac)(ceb)) = ac

(3)

= (1) + (2) d(cxb)(d(ab))x = ac

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(4)

= (1)[a, c/cx, b, x/a(cx)] cx(a(cx))(cx(cx))(cx(a(cx))(a(cx))) = a(cx)

(5)

= (3)[d/cx(a(cx)), b/cx] cx(a(cx))(cx(cx))(cx(a(cx))(a(cx)))x = ac

(6)

= (4) + (5) a(cx)x = ac

(7)

= (3)[x/ab] d(c(ab)b)(d(ab))(ab) = ac

(8)

= (6)[a/d(c(ab)b), c/d, x/ab] d(c(ab)b)(d(ab))(ab) = d(c(ab)b)d

(9)

= (7) + (8) d(c(ab)b)d = ac

(10)

= (9)[d/eb(ec)] eb(ec)(c(ab)b)(eb(ec)) = ac

(11)

= (1)[a/e, x/ab] eb(ec)(c(ab)b) = ab

(12)

= (11) + (10) ab(eb(ec)) = ac

(13)

by (12) ab(eb(ec))(ec) = ac(ec)

(14)

= (6) + (13) ab(eb) = ac(ec)

(15)

= (14)[e/a] ab(ab) = ac(ac)

(16)

= (1)[a, c/a, b, x/aa] a(aa)(aa)(a(aa)(aa)) = aa 147

(17)

= (15)[a/a(aa), b/aa, c/b] a(aa)(aa)(a(aa)(aa)) = a(aa)b(a(aa)b)

(18)

(16) + (17) a(aa)b(a(aa)b) = aa

(19)

by (18) a(aa)b(a(aa)b)(bxb) = aa(bxb)

(20)

= (19) + (1)[a/a(aa), c/b] aa(bxb) = x

(21)

by (20) aa(bxb)b = xb

(22)

= (6) + (21) aa(bx) = xb

(23)

= (22)[b/x] aa(xx) = xx

(24)

by (23) aa(aa)(aa(aa))(aa(xx)(aa)) = aa

(25)

= (1)[a, b, c/aa, x/xx] aa(aa)(aa(aa))(aa(xx)(aa)) = xx

(26)

= (24) + (25) aa = bb

(df)

aa = 0

(27)

= (22) + (df ) 0(ab) = ba

(28)

= (1) + [b, c/a] + (df ) 0(axa) = x 148

(29)

= (28) + (27) a(ax) = x

(30)

= (29) + (27) 0(0a) = a

(∗1)

= (30) + (27) a0 = a

(31)

= (1)[x/cb] ab(ac)(c(cb)b) = cb

(∗2)

= (31) + (29) + (∗1) ab(ac) = cb

References [1] K. Iseki, An algebra related with a propositional calculus, Proc. Japan Acad., 42, 1966, pp. 26-29. [2] Jacek K. Kabzi´ nski, Abelian group and identity connective, Bulletin of the Section of Logic Polish Academy of Sciences, 22, 1993, pp. 66-71. [3] Jacek K. Kabzi´ nski, Basic Properities of the Equivalence, Studia Logica, 41, 1982, pp. 17-40. [4] Roman Suszko, Equational logic and theories in sentential languages, Colloquium Mathematicum, 29, 1974, pp. 19-23.

Department of Logic Jagiellonian University Grodzka 52 31–044 Krak´ow Poland

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