Survey 1979: Equational Logic - Department of Mathematics ...

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2. xy = yx 3. x + (y+z)-- (x+y) + z 4. x(yz) = (xy)x 5. x(y+z) = xy +xz 6. x y+z = xYx z 7. (xy) z= xZy z 8. (xy)Z = x(yZ) 9. x'l =x 10. xl=x 11. lX=l. WALTER TAYLOR 17 (Tarski - see [286] .) Tarski has called this the "high school identity problem (with unit)." (Incidentally, Martin [286] has remarked that it follows easily from the methods of Richardson [377] that Eq(oo, 1 ,x+y,xy,xY) = Eq(R +, 1 ,x+y,xy,xY), where R + is the set of non-negative real numbers.) We could go on and on with interesting examples (see e.g. [4], [104], [135], [ 153], [ 188], [265], and [408] ), but we will stop here. In place of finding a base of a given A, one can often be content with the knowledge that a finite 0 exists or does not exist, as the case may be: this is the idea of the next section. (Although sometimes explicit - but complicated - bases are found in the researches reported in õ 9 and õ10. The methods of õ 14.5 and õ15 also sometimes lead to finding 20 and A as in this section.) The reverse problem, of finding a generic A for a given 20' is less well defined. As remarked above, A always exists, but finding a "known" (i.e. familiar) or simple generic algebra can be very elusive, e.g. for modular lattices. (The problem of "simply" describing a free algebra is really a word problem - see õ 12 below - and the word prcblem for free modular lattices is not solvable.) 9. FinRely based theories. We say that an equational theory 2 is nitely based iff there exists a finite set 20 of axioms for 2. (The definitions in õ7 should make it clear that this is an equivalence-invariant property of all 2 which have finitely many operations.) Evidently many familiar theories are finitely based - groups, Boolean algebras, rings, lattices, etc.; see also the various examples in õ8. Here we list some
18 EQUATIONAL LOGIC algebras A with Eq (A} known to befinitely based: 9.1. Any two-element algebra (Lyndon [265] ). (But Cf. 9.16 below.) 9.2. Any finite group (Oates and Powell [336] ). 9.3. Any commutative semigroup (Perkins [349] ). (In other words, every variety of commutative semigroups is finitely based. This is also proved in [ 124] .) Also, any 3-element semigroup [349]. (Cf. 13.5 below.) 9.4. Any idempotent semigroup (Fennemore [137], Biryukov [54], Gerhard [ 152] ). (A semigroup is idempotent iff it obeys the law x 2 = x.) 9.5. Any finite, simple, 2-generated quasigroup (McKenzie [297] ). 9.6. Any finite ring (Kruse [250], Lvov [263]). 9.7. The ring M2(k ) of 2 X 2 matrices over a field k of characteristic 0. (Razmyslov [374] .) (Cf. 8.4 above.) (For n > 3, this is open.) 9.8. Any nilpotent ring; any commutative ring (Bang and Mandelberg [37] ). 9.9. Any finite (non-associative) ring without zero-divisors (Lvov [264]). 9.10. Any finite lattice (possibly with operators)(McKenzie [291]). (Answering Problem 45 in Gr/itzer's book [ 163] .) More generally: 9.11. Any finite algebra which generates a congruence-distributive variety (see {} 15 below) (Baker [ 17] - see also [277], [426] and [226] ). The special case (of 9.10 - 9.11) of primal algebras was known much earlier (Rosenbloom [379], Yaqub [444]; also Yablonskit' in the mid-fifties - see [ 319] ). 9.12. Any finite simple algebra with no proper subalgebras except one-element subalgebras which generates a congruence-permutable variety (McKenzie [299] ). 9.13. If V has only finitely many subdirectly irreducible algebras, all of them are finite, and V has definable principal congruence relations, then V is finitely based. As a corollary, if V is a locally finite variety and there exist A1,...,A k 6 V so that every n 1 n2 nk finite A6 V is isomorphic to some A 1 A 2 'ø' Ak, then V is finitely based. (McKenzie [300].) Thus the para-primal varieties of Clark and Krauss are finitely based. 9.14. Any finite ©-product of finitely based theories is finitely based (see [420], pages 357-358]; [424, pages 266-267] for © - which corresponds to taking the product of the algebraic theories described in {}7). Pursuing the analogy at the end of
Page 1 and 2: HOUSTON JOURNAL OF MATHEMATICS Surv
Page 3 and 4: EQUATIONAL LOGIC Walter Taylor This
Page 5 and 6: EQUATIONAL LOGIC CONTENTS 1. Early
Page 7 and 8: 2 EQUATIONAL LOGIC 2. The existence
Page 9 and 10: 4 EQUATIONAL LOGIC A are taken as v
Page 11 and 12: 6 EQUATIONAL LOGIC is closed under
Page 13 and 14: 8 EQUATIONAL LOGIC model theory, an
Page 15 and 16: 10 EQUATIONAL LOGIC completeness th
Page 17 and 18: 12 EQUATIONAL LOGIC algebras do not
Page 19 and 20: 14 EQUATIONAL LOGIC groups with a s
Page 21: 16 EQUATIONAL LOGIC (Martin [286, p
Page 25 and 26: 20 EQUATIONAL LOGIC 9.18. Some vari
Page 27 and 28: 22 EQUATIONAL LOGIC some algebras o
Page 29 and 30: 24 EQUATIONAL LOGIC is an interval
Page 31 and 32: 26 EQUATIONAL LOGIC The second meth
Page 33 and 34: 28 EQUATIONAL LOGIC for arbitrary f
Page 35 and 36: 30 EQUATIONAL LOGIC The theory Z 1
Page 37 and 38: 32 EQUATIONAL LOGIC conversely, eve
Page 39 and 40: 34 EQUATIONAL LOGIC x=y Boolean alc
Page 41 and 42: 36 EQUATIONAL LOGIC dual covers: on
Page 43 and 44: 38 EQUATIONAL LOGIC (Some prelimina
Page 45 and 46: 40 EQUATIONAL LOGIC semilattices (c
Page 47 and 48: 42 EQUATIONAL LOGIC a congruence on
Page 49 and 50: 44 EQUATIONAL LOGIC compactificatio
Page 51 and 52: 46 EQUATIONAL LOGIC precise definit
Page 53 and 54: 48 EQUATIONAL LOGIC 16. Connections
Page 55 and 56: 50 EQUATIONAL LOGIC 17.3. See e.g.
Page 58 and 59: REFERENCES This is not really a com
Page 60 and 61: 52.* __ , Current trends in algebra
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Page 66 and 67: 216. Jones, G. T., Pseudocomplement
Page 68 and 69: 274. 275. 276. 277. Ferrara (7), 14
Page 70 and 71: 324. The lattice of equational clas
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Soc., 23(1976), A-93. {16.7) 379. R
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15(1974), 174-177. (13} 431.*Trakht
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72 , Certain algorithmic problems f
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74 161-171. Fajtlowicz, S. and E. M
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76 Goetz, A. and C. Ryll-Nardzewski
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78 sets and general algebra, Acta.
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8O Newman, M. H. A., A characteriza
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82 __, Trellis theory, Memoirs Amer
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The Department of Mathematics and t
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