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

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WALTER TAYLOR 33 (All theories intended as extensions of 23 = {x(yz) = (xy)z} .) 13.5. Forcommutativesemigroups, see [349], [179], [325],[388],and [78]. Perkins proved that every commutative semigroup theory is finitely based (cf. 9.3 above), and hence this lattice is countable. And so it cannot contain IIoo, but it does contain every IIm (Burris-Nelson [78]) and hence obeys no special lattice laws; Schwabauer had earlier proved [388] that the lattice of commutative semigroup theories is nonmodular. For semigroups with zero, consult [324], [83], and with unit [179]. For related work see [344], [350], [351]. 13.6. For 23 = ((xy)z = x(yz), x 2= x} ("idempotent semigroups"), A(23) has been completely described by Biryukov [54], Fennemore [137] and Gerhard [152]. In this picture, the diamond pattern repeats indefinitely in the obvious way: xy=x xyz = xz y xy = xyx xyz = xyxz xyz = xyzxz xyz = xyzxzyx xy=y xzy = zxy xy = yxy xyz = xzyz xyz = xzxyz xyz = xyxzxyz (all [hexries are intended as extensions of 23). Note that this lattice is countable, distributive and of width three. The situation is very different for 23' = (x(yz) = (xy)z, x 2 = x3} ß Burris and Nelson [79] proved that Iloo C_ 13.7. For 23 the equational theory of distributive lattices with pseudocomplementation, A(23) is an infinite chain:
34 EQUATIONAL LOGIC x=y Boolean alcjebro Stone o Icjebra (Lee [256]; see also [165]). A similar result holds for one-dimensional polyadic algebras (Monk [ 315 ] ). 13.8. If 23 = (xy--yx, (xy)(zw)= (xz)(yw)), then A(23) is uncountable, even above 23 t (x 2 = y2). But A(23 t (x 2 = x)) is countable (and explicitly described). See [213]. 13.9. For Heyting algebras, consult Day [102]; and for the closely related Brouwerian algebras, see K6hler [249]. Also closely related are interior algebras;their varieties correspond to modal logic extensions of Lewis' S4 (W. J. Blok, thesis). 13.10. For lattice-ordered groups see Marti'nez [288]; their lattice has a surprising superficial similarity to that of Heyting algebras (above). Holland showed [ 188] that every proper extension of the theory of lattice-ordered groups contains (an identity equivalent to the implication) (1 a)&(1 b):*abb2a 2, and thus this lattice has a unique atom. See Scrimger [389] for a study of the theories just below the theory of Abelian lattice-ordered groups. 13.11. A(L) has been extensively studied for L = lattice theory (see e.g. [93 ] ). It is a distributive lattice which contains, in part
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 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: 32 EQUATIONAL LOGIC conversely, eve
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
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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The Department of Mathematics and t
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