ON THE UNIQUE AXIOM OF BCII CLASS

Jacek K. Kabziński
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
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(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
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(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. Kabziński, Abelian group and identity connective, Bulletin of the Section of Logic Polish Academy of Sciences, 22, 1993,
pp. 66-71.
[3] Jacek K. Kabziński, 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 Kraków
Poland
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