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

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WALTER TAYLOR 19 õ7, this easy result corresponds to the fact that the product of two finitely presented groups is finitely presented. 9.15. Recently Murskii has proved [319] that "almost all" finite algebras have a finite base for their identities (i.e., for fixed type, the fraction of such algebras among all algebras of power k approaches 1 as k - oo _ or even, for fixed k, as the number of operations approaches oo). (The unary case is easy - all are finitely based; in the non-unary case he in fact proves much more: almost all are quasi-primal - cf. õ15 below, and also 9.11 and 10.7.) For some further remarks about finite algebras with finite bases, consult [235]. We now turn to equational theories which are not finitely based. Of course it is almost trivial to construct such theories using infinitely many operations Ft(t 6 T), even some which are equivalent to the (finitely based!) theory with no operations. As G. Bergman pointed out,non-finitely based theories with finite T arise almost automatically if we consider a semigroup S which is finitely generated (say by F C_S), but not finitely related. Our theory can be taken to have unary operations for f F and laws flf2 "' fk x= fk+l "' fs x whenever fl '" fk = fk+l "' fs in S. Some more interesting research has centered on finding less obvious, but more important, examples of theories and algebras which have T finite and are still not finitely based: 9.16. The algebra with universe (0,1,2) and binary operation: (Murski]' [ 318 ] 9.17. The o I ; 0 0 0 0 I 0 0 I , following Lyndon [266] ). six-element semigroup 0 (with ordinary matrix multiplication). (Perkins [349].) Earlier Austin [ 11 ] gave some other varieties of semigroups which are not finitely based. C. C. Edmunds has recently shown that six is as small as possible for a semigroup with zero and unit.
20 EQUATIONAL LOGIC 9.18. Some varieties of groups (Ol'shanskii [337], Vaughan-Lee [434] )(cf. 13.3 below for a more complete discussion). The existence of such varieties of groups was open for a long time. 9.19. The infinite lattice (McKenzie [ 291 ] ). 9.20. The algebras (2,+,-) (Martin [286, page 210] [287]) and (2,x y) [286, page 211]. See 8.10 and 8.12 for the definitions and some comparisons. Note also that (co,+,') is obviously finitely based (cf. 8.1 ). Also cf. õ 12 below. 9.21. The algebra (co,x+y,xy,xY) (Martin [286, page 118] ). (Cf. 8.11.) This is a negative solution to one version of Tarski's "high school identities problem"-he described a set of 8 familiar identities (namely the first 8 of Problem 2 of õ8), and asked if these formed an equational base. For another version of this problem, see Problem 2 below and Problem 2 of õ 8. 9.22. Any lattice-ordered ring which is an ordered field (and all of these have the same equational theory) (Isbell [ 199] ). (An infinite basis is indirectly described [loc. cit. ].) 9.23. The variety of representable relation algebras (Monk [312] ) and for n 3 that of representable cylindric algebras of dimension n (Monk [314]). (Roughly speaking, cylindric algebras are to full logic what Boolean algebras are to logic without quantifiers"forall,""ther½ exists'.' Relation algebras are intermediate in strength.) Representable algebras (of either type) have a very natural semantic definition; the definition of the entire class of cylindric or relation algebras amounts to selecting a (necessarily rather arbitrary, no matter how utilitarian) finite subset of the equational theory of representable algebras. Monk's results indicate that there is really no natural finite set of axioms.
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: 18 EQUATIONAL LOGIC algebras A with
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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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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