DEMONSTRATIO MATHEMATICA

Vol. XVII No 4 1984

M. Gebel, J. Lang, M. Stem

A GENERAL ISOMORPHISM THEOREM FOR UNIVERSAL ALGEBRAS

1. Introduction The idea of giving a common generalization of both group theoretic isomorphism theorems can be found in [l] . Using the theory of standard ideals in lattices (cf. [3])• a simi- lar generalization oan be proved for both lattice theoretic isomorphism theorems (cf. [4])* In [4] the corresponding problec was posed for universal algebras. It is the aim of this paper to answer this problem affirmatively. We prove a general isomorphism theorem for universal algebras (s. Section 3) from which both the First and the Second Isomorphism Theorems as formulated in [2, Theo- rem 2, p.58 and Theorem 4, p.62] can be derived.

2. Preliminaries Throughout this paper we adopt the notation of [2] . All background material can likewise be found in [2] . In order to make the presentation more self-contained, we list here the most important notions and results needed in the sequel. A universal algebra Cft is a pair <(Aj ?)> where A is a nonvoid set and F is a family of finitary operations on A. In this paper, we restrict ourselves to a finite nonvoid F. Let 01 be an algebra, X a subalgebra of C/L and 0 a con- gruence relation of CM . Then [fl]e denotes the 0-closure of X , that is, a0,...,an e [b]0 (Bq,...,an ^ e A) if

and only if ai = b^S) (0 = i

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For an algebra Ul, a subalgebra £ of CM and a congruence relation 0 of (ft, denotes the restriction of 8 to B. For an a e A we mean by [a] e the congruence class to which a belongs* 3y Crt./9 we mean the quotient algebra . We shall need the Homomorphism Theorem (cf. [2, Theo- rem 1, P.57]). Let Ul and £ be algebras and

Then < [B] 8 J F> is also a subalgebra of VI . Finally, we shall need the following notation whioh is introduced in order to obtain a complete description of the congruence relations of the quotient algebra Crt/8. To this 33d let binary relation 0/9 on A/@ by [a] 0 = [b] 0 (4>/®) if and only if

a = b(/3 is a congruence relation of C/t/®.

3. A general isomorphism theorem for universal algebras In this section we prove the following Theorem. Let C/t be an algebra and £ be a sub- algebra of C/t. Let further 0 , be congruence relaticns of C/t and assume that 0 S . Then

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<([B]0/e[B]0)/(«[B]e/e[Bj0)j p>a

a <[B]O/«[b](|)» ?>.

Proof. Consider the mapping

which is effeoted by (We[B]8)«p- W

where b c [B] 8. By definition q> is onto. Moreover cp is well-defined. Namely, if

[bl]®[B]0 » [b2]0[B]e

with b^ ,bg e [B]e, then b., s 8X1(3 therefor® 5 2 b2^[B]d>^ because of

[M^BJD s [b2] ^ [B] *

Furthermore,

" MM^B]* [bn^-l] ^[b]«^ *

- bn.p-1^ 9[B]8^' Thus f is a homomorphism. Let v denote the congruence relation induced by (p. Then we have by the Homomorphism Theorem (cf. Seotion 2) that

909 4 M. Gebel, J. Lang, M* Stern

< ( [B] e/0 fBje )/vt ?> s < [B]^ ? > where an isomorphism is given by

with b e [b] 8. Therefore

8 b 0 N [B]8 » [ 2] [B]8<*> holds if and only if

{[bl]9[B]eW« <[b2]0[B]8>

wi;h ,b2 s [b]8. From the definition of

N^CB]« • [b2]$[B]- This is equivalent to

and henoe also to

(«) b, = b2(«[B]e)

8 ino e b^bg E [B] 8. From (*) and (**) we get that

[blJ9[B]9 s [b2]®[B]8<^ holds if and only if

b1 s b2([B]0)-

V is a binary relation on [b]8/8|-bjq. Por this binary rela- tion we may use the notation ^ « ^[Bjs^fBle Section 2). According to [2, Lemma 1, p.59] (s. also Section 2) «y is even a congruence relation on - 910 - General isomorphism theoren 5

[B]e/e[B]e. This implies the assertion of the theorem* Bote that this theorem provides an affirmative answer to the problem posed in 4 • Corollary [1] (First Isomorphism Theorem, cf. [2» Theorem 2, p.58]). Let C/t be an algtbra, £ a subalgebra of 01 , and a congruenoe relation of CI. Then

Fro of. Putting 8 • u> in the preceding theorem (w denotes the least congruenoe relation) we get

Observing that [B] w = B, = u>B and •[B]t*/<«>[B]W = *B/«B » = B it follows that

B;F> 8 <[B]. This proves the corollary* Corollary 2 (Seoond Isomorphism Theorem, of. [2, Theorem 4, p.62]). Let C/t be an algebra, let e , 0 be oongruenoe relations of VI and assume that 8£* Then Ot/OS (Crt/e)/(0/9).

Proof. Under our assumptions, we have B C[B] 9 £ [B] S A. Putting B * A in the preceding theorem the asser- tion follows immediately. Remark (added in proof)» Professor G. Kalmbaoh (University of Ulm, GFR) has kindly called our attention to the fact that Professor G. Banasohewski (McMaster University, Hamilton, Ontario, Canada) has found another version of a proof of the isomorphism theorem treated in the present note. To the best of our knowledge, Banasohewski's proof has not been published in a journal. - 911 - 6 M. Gebel, J. Lang, M. Stern

Acknowledgments. The third author gratefully acknowledges support by the natural Sic ie no es and Engineering Sa search Council of Canada, Grant 214-1518.

REFERENCES

[1] Frieds Absztrakt algebra elemi uton. Budapest 1575. [2] G« Q r a t a e r s Universal Algebra. Princeton, New Jersey, 1968. [3] G. Gratzer, B.T. Schmidt t Standard Ideals In lattices, Acta Math. Aoad. Sei. Hung. 12 (1961)

[4] M. .".tern: An isomorphism theorem for standard ideal,/ in lattices, Algebra Universalis (in print).

Auth ors' ad dra« a« s j «.Gebell GORKIJ UNIVERSITY, CHARKOW, U.S.S.R. J.Lsag/M.Stern: MARTIN-LUTHER-UNIVERSITÄT. SECTION MATHEMATIK, DLR-402 HALLE Received June 14, 1983«

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