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

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WALTER TAYLOR 47 Every finite algebra in HSP A is uniquely a product of subalgebras of A. Many of the equivalence-invariants of these varieties (e.g. the fine spectrum, CEP, AP, Con(V) - see õ14) are relatively easy to evaluate. See Pixley [362] and Quackenbush [369] for details and further references - the notion goes back essentially to Pixley, building on work of Foster and Rosenbloom. For infinite analogs of primal algebras, see Tulipani [433] and Iwanik [202]. For congruence-distributivity cf. also 9.11 and 14.7. Clark and Krauss [89] [90] have given a remarkable theory of para primal algebras, a kind of non-distributive generalization of quasiprimal algebras, combining ideas of quasiprimality and linear algebra. Also see [371] [300] [173]; cf. 9.13 above. Pixley and Wille gave an algorithm ([363] [442]; also see [420, Theorem 5.1]) to convert every identity on the congruence lattice (in ^, v, and o) into a Malcev condition. Which of these conditions are "new" remains an open question. For instance, Nation proved [322] that for certain lattice laws X which do not imply the modular law, the following holds: if all congruence lattices of algebras in a variety V obey X, then they all obey the modular law. Very recently S. V. Polin has proved that Nation's result fails for some non-trivial lattice law X. Non-modular "congruence varieties" are extensively investigated in a forthcoming paper of A. Day and R. Freese. Also see [103], [224] or [298] for further discussion of the state of affair just prior to Polin's result. Also see B. Jdnsson's appendix to the forthcoming new edition of [ 163 ]. Another Malcev-definable property of varieties V which has received wide attention is that Fv(n) Fv(m). (See e.g. Marczewski [282], references given there, and various other articles in the same volume of Colloquium Mathematicum.) For fixed no, the set of numbers {n C co: Fv(n) Fv(n0)) is always an arithmetic progression, and any progression can occur (S'wierczkowski, et al.). If Fv(n ) - Fv(m ) with m =/= n, then V has no non-trivial finite algebras (Jdnsson and Tarski [231]) (cf. 14.1 ). (Also see Clark [87] .) See Csfikiny [95] for a collection of properties of varieties resembling, but more general than, Malcev conditions. A nice special example is in Klukovits [247].
48 EQUATIONAL LOGIC 16. Connections with topology. If A = (A,Ft)tC T is any algebra and T is any topology on A such that every F t' A nt-* A is continuous (in the nt-fold topological product of T), then we say that A = (A,T,Ft)tC T is a topological algebra. The space (A,T) is not at all independent from the equational theory of A, but rather the two seem to influence each other quite strongly. This influence is poorly understood, but many interesting examples are known. 16.1. No sphere except S 0, S 1 , S 3, S 7 can be an "H-space," i.e., can obey ex = x = xe for binary multiplication and constant e (Adams [2]), and S 7 cannot also obey the associative law (James [204] ). 16.2. The space of a topological group must be homogeneous, with Abelian fundamental group, and if compact and uncountable, of power > 2 q0. (All of these facts are essentially well known.) 16.3. The space of a topological Boolean algebra, if compact, must be a power 2 n for 2 a 2-element space. (Kaplansky [237] .) 16.4. The space of topological semilattice has zero homotopy in each of its components in each dimension (Taylor [429]), and if compact, connected and finite-dimensional, cannot be homogeneous (Lawson and Madison [253] ). 16.5. If V is defined by the equations (*) of 14.3 above, then it is not hard to check that the topological algebras in V have "square" universe (as in Evans [ 118] ), i.e. each is homeomorphic to the square of some other space. 16.6. If V is the product U W of two varieties (see [424], õ0]), then a topological algebra in V with product-indecomposable space must be either in U or W (this can be seen fairly easily from the methods outlined in [420, pages 357-358] or [424, pages 265-267] ). 16.7. If a compact connected topological algebra obeys the law (xy)(yz) = xz, then it also obeys the law xy = uv. (Bednarek and Wallace [38] .) It is an old problem of A.D. Wallace (see [378] ) whether the "skew-associative" law x(yz) = (zx)y can hold on the unit interval without the associative law holding as well.
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
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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
Page 25 and 26: 20 EQUATIONAL LOGIC 9.18. Some vari
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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: 46 EQUATIONAL LOGIC precise definit
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
Page 62 and 63: Universalis, 2(1972), 317-323. { 12
Page 64 and 65: Math. Bull., 18(1975), 427-429. 161
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
Page 72 and 73: Soc., 23(1976), A-93. {16.7) 379. R
Page 74: 15(1974), 174-177. (13} 431.*Trakht
Page 77 and 78: 72 , Certain algorithmic problems f
Page 79 and 80: 74 161-171. Fajtlowicz, S. and E. M
Page 81 and 82: 76 Goetz, A. and C. Ryll-Nardzewski
Page 83 and 84: 78 sets and general algebra, Acta.
Page 85 and 86: 8O Newman, M. H. A., A characteriza
Page 87 and 88: 82 __, Trellis theory, Memoirs Amer
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