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

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WALTER TAYLOR forms of mathematics from the Theorem of Pythagoras onward). While the maturity and value of mathematical logic are unquestioned nowadays, we hope the reader will also gain an insight into the present-day vigor (if not yet maturity) of general algebra. This subject has been clouded by a skepticism ranging from Marczewski's [283] sympathetic warning: [In subjects like general topology and general algebra] it is easy to get stranded in trivial topics, and caught in the net of overdetailed conditions, of futile generalizations. to the outright malediction: "nobody should specialize in it" ([184], [64]). This injunction is certainly out of date (if indeed it ever was valid), and we hope this survey will be adequate evidence of the successes of specialists in universal algebra. And perhaps this survey will help dispel another (closely related) myth, which is epitomized by Baer's remark [16, page 286], "The acid test for [a wide variety of methods in universal algebra] will always be found in the theory of groups." Many interesting results and ideas here either collapse completely or become hopelessly complicated when applied to groups; but there is no lack of interesting classes of algebras defined by equations to which the theories may be applied, as we shall see. And this again is one of the attractions of the subject. The writing of this survey was supported, in part, at various times, by the University of Colorado, the Australian-American Educational Foundation and the National Science Foundation. An early ancestor of this survey was a report of the Australian S. R. I. lectures which appeared in the proceedings of the Szeged Universal Algebra conference of 1975. Although we believe the matehal unfolds rather naturally in the order we present it, only õ õ 1,2, 3, 5 are essential to read first.
EQUATIONAL LOGIC CONTENTS 1. Early history and definitions ................................................... 1 2. The existence of free algebras .................................................. 2 3. Equationally defined classes ................................................... 3 4. Generation of varieties and subdirect representation ................................. 6 5. Equational theories .......................................................... 8 6. Examples of . ............................................................. 9 7. Equivalent varieties .......................................................... 11 8. Bases and generic algebras ..................................................... 14 9. Finitely based theories ....................................................... !7 10. One-based theories .......................................................... 21 11. Irredundant bases ........................................................... 23 12. Decidability questions ........................................................ 24 13. The lattice of equational theories ............................................... 29 MODEL THEORETIC QUESTIONS ............................................. 37 14. Some further invariants of a variety ............................................. 37 15. Malcev conditions and congruence identities ....................................... 45 16. Connections with topology .................................................... 48 17. Miscellaneous .............................................................. 49 References ................................................................ 53
Page 1 and 2: HOUSTON JOURNAL OF MATHEMATICS Surv
Page 3: EQUATIONAL LOGIC Walter Taylor This
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 and 22: 16 EQUATIONAL LOGIC (Martin [286, p
Page 23 and 24: 18 EQUATIONAL LOGIC algebras A with
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
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50 EQUATIONAL LOGIC 17.3. See e.g.
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REFERENCES This is not really a com
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52.* __ , Current trends in algebra
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Universalis, 2(1972), 317-323. { 12
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Math. Bull., 18(1975), 427-429. 161
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216. Jones, G. T., Pseudocomplement
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274. 275. 276. 277. Ferrara (7), 14
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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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