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

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HOUSTON JOURNAL OF MATHEMATICS D. G. Bourgin Department of Mathematics University of Houston Houston, Texas 77004 Bjarni J6nsson Department of Mathematics P.O. Box 1541, Station B Vanderbilt University Nashville, Tennessee 37205 Andrzej Lelek Department of Mathematics University of Houston Houston, Texas 77004 J. S. Mac Nerney Department of Mathematics University of Houston Houston, Texas 77004 Editors Managing Editor Gordon G. Johnson .Department of Mathematics University of Houston Houston, Texas 77004 J. W. Neuberger Department of Mathematics North Texas State University Denton, Texas 76203 B. H. Neumann Department of Mathematics Institute of Advanced Studies Australian National University P.O. Box 4 Canberra, ACT 2600 Jtirgen Schmidt Department of Mathematics University of Houston Houston, Texas 77004 Associate Managing Editors Jutta Hausen Johnny A. Johnson Published quarterly by the Department of Mathematics, University of Houston, Houston, Texas 77004. The price per volume is $35.00; outside of the United States and Canada, $40.00. Each volume consists of four quarterly issues. Subscription orders should be addressed to the Managing Editor. Copyright ¸ 1979 by the University of Houston, Department of Mathe- matics, Houston, Texas. Composed by Kit Hensley and printed by the University of Houston in Houston, Texas.
EQUATIONAL LOGIC Walter Taylor This is a survey of existing work of many authors in equational logic or varieties of algebras. Our primary interest is in equations for general algebraic systems, and we will not report in detail on equations in special systems (e.g., fields, where equations began). As a branch or "fragment" of general first order logic, this subject has two aspects, one focusing attention on the formal expressions (in this case, the equations), and the other focusing on the models of these equations. We give slightly more attention to the first of these, focusing, until õ 13, on sets of equations. This survey owes much to an expository article of Tarski [413] in 1968, and to some unpublished notes of D. Pigozzi (ca. 1970). Our exposition will be self-contained for the general mathematician, the only special prerequisite being a rudimentary understanding of the term "decidable." We include no proofs. This survey originated in a series of talks at the 1975 Summer Research Institute of the Australian Mathematical Society. Some valuable suggestions about this article were made by G. Bergman, W. J. Blok, S. Comer, B. Cskfiny, B. Davey, A. Day, G. Gr//tzer, W. Hodges, B. J6nsson, R. McKenzie, G. McNulty, J. Mycielski, E. A. Palyutin, D. Pigozzi, A. Pixley, and A.D. Tat'manov. Our objective is to make more mathematicians aware of this subject and to provide a readable introduction to its examples, its theorems, and what they mean in a fairly broad context. At the same time we hope to provide a reasonably complete survey of the literature which will be helpful to specialists. For these reasons (and others) we have omitted all proofs, and so perhaps we should warn the reader of one aspect of the subject on which we have not commented in detail: which of these results were most difficult to prove. But here we may mention one of the attractions of equational logic: there are hard and interesting theorems which are very easy to state (a property of all attractive
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Page 19 and 20: 14 EQUATIONAL LOGIC groups with a s
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Page 23 and 24: 18 EQUATIONAL LOGIC algebras A with
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Page 29 and 30: 24 EQUATIONAL LOGIC is an interval
Page 31 and 32: 26 EQUATIONAL LOGIC The second meth
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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
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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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