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Definable principal congruences and solvability

2009
*
Annals of Pure and Applied Logic
*

We prove that in a locally finite variety that has

doi:10.1016/j.apal.2008.09.019
fatcat:6tbmha4mevdtbdqb74kd2hlnga
*definable**principal**congruences*(DPC),*solvable**congruences*are nilpotent,*and*strongly*solvable**congruences*are strongly abelian. ... is a strongly*solvable**congruence*of A, then β is strongly abelian. (3) If V(A) is*congruence*modular*and*A is*solvable*, then A can be decomposed as a direct product of nilpotent algebras of prime power ... T043671*and*T043034*and*the NSERC of Canada. ...##
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Inherently nonfinitely based solvable algebras

1994
*
Canadian mathematical bulletin
*

On the other hand, we show by example that it is possible for an inherently nonfinitely based algebra to generate a strongly

doi:10.4153/cmb-1994-074-6
fatcat:662ytnrhirgz7dfch2426chgzi
*solvable*variety. * Research supported by a grant from NSERC. 1 ... W has*definable**principal**congruences**and*, in fact, φ is a formula which*defines**principal**congruences*in W. For each i ∈ I*and*each p j occuring in π i there is a Klukovits term g ij for p j . ... Proof: Let V be a Hamiltonian variety with*definable**principal**congruences*. ...##
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Page 960 of Mathematical Reviews Vol. 50, Issue 4
[page]

1975
*
Mathematical Reviews
*

system of polynomial equations, each depending on x,

*and*with constants in , is*solvable*in % provided it is finitely*solvable*in 2.” ... The author*defines*the concepts of operation, groupoid, homo- morphism*and*isomorphism,*congruence*, groupoid of subsets of a given groupoid, subgroupoid, generating set, direct product, commutativity, ...##
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Ray class field extensions of real quadratic fields and solvability of congruences

1985
*
Journal of Number Theory
*

An explicit description is given of a process using classical class field theory to generate

doi:10.1016/0022-314x(85)90021-6
fatcat:grktksux2rgpjecaldwzatty2a
*solvability*criteria for a class of fourth degree*congruences*. ... The method involves finding generators*and*determining conductors for relatively quadratic extensions of a real quadratic base held. Several examples are given. '(J 1985 Academic Press. Inc. ... However, all the*congruences*are*solvable*for the same set of primes q. (1 -l/N(p)), the generalized Euler phi function, is*defined*in terms of absolute norm N of the ideal f (the finite part off) (cf. ...##
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Commutator theory for racks and quandles
[article]

2020
*
arXiv
*
pre-print

*Congruence*properties such as abelianness

*and*centrality are reflected by the corresponding relative displacement groups,

*and*so do the global properties,

*solvability*

*and*nilpotence. ... We adapt the commutator theory of universal algebra to the particular setting of racks

*and*quandles, exploiting a Galois connection between

*congruences*

*and*certain normal subgroups of the displacement ... Polynomial equivalence preserves all properties

*defined*by polynomial operations, such as

*congruences*, the centralizing relation C(α, β; δ),

*and*subsequently the notions of abelianness,

*solvability*, etc ...

##
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Residual finiteness growths of Lamplighter groups
[article]

2019
*
arXiv
*
pre-print

In our proof, we quantify a

arXiv:1909.03535v2
fatcat:lcy3pjohpzhxfnimlfxrhbrjgi
*congruence*subgroup property for lamplighter groups. ... We then improve on the best known upper*and*lower bounds for lamplighter groups. Notably, any lamplighter group has super-linear residual finiteness growth. ... Acknowledgements We are grateful to Ahmed Bou-Rabee, Rachel Skipper,*and*Daniel Studenmund for giving us comments*and*corrections on an earlier draft. ...##
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The commutator in equivalential algebras and Fregean varieties

2011
*
Algebra Universalis
*

In this paper we give a full characterization of the commutator for equivalential algebras

doi:10.1007/s00012-011-0133-4
fatcat:uchepwrlm5gyxjnjp7mecolrqq
*and**solvable*Fregean varieties. ... A class K of algebras with a distinguished constant term 0 is called Fregean if*congruences*of algebras in K are uniquely determined by their 0-cosets*and*Θ A (0, a) = Θ A (0, b) implies a = b for all ... Since A is*solvable**and*belongs to a*congruence*modular variety, the variety V(A) generated by A is*solvable*. ...##
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Mal'cev classes of left-quasigroups and Quandles
[article]

2021
*
arXiv
*
pre-print

According to the commutator theory developed in [FM87]

arXiv:2004.05368v2
fatcat:c6iomhvhangghpzo3yng4gvtiu
*and*adapted to racks in [BS19b] we can*define*abelianess*and*centrality for*congruences*of general algebras*and*consequently nilpotence*and**solvability*... by using a special chain of*congruences**defined*in analogy with the derived series*and*the lower central series of groups, using the commutator between*congruences*as*defined*in [FM87] (we denote the ...##
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Congruences in Algebraic Number Fields Involving Sums of Similar Powers

1956
*
Transactions of the American Mathematical Society
*

In the final section ( §5) estimates for Q,(p) are obtained (Theorems 6

doi:10.2307/1992888
fatcat:qx3doactrjcmte55mihii4sopa
*and*7)*and**solvability*criteria for the*congruence*(1.1) are deduced (Theorems 8*and*9). ... We place rn = f0x~" (O^ragA) so that £ = £\j*and**define*further We denote by x a fixed, primitive &th power character ( mod P)*and*let Xo represent the*principal*character (mod P). ...##
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Congruences in algebraic number fields involving sums of similar powers

1956
*
Transactions of the American Mathematical Society
*

In the final section ( §5) estimates for Q,(p) are obtained (Theorems 6

doi:10.1090/s0002-9947-1956-0084524-3
fatcat:ynkf7tnoujan7eb2g6dnhvfpxe
*and*7)*and**solvability*criteria for the*congruence*(1.1) are deduced (Theorems 8*and*9). ... We place rn = f0x~" (O^ragA) so that £ = £\j*and**define*further We denote by x a fixed, primitive &th power character ( mod P)*and*let Xo represent the*principal*character (mod P). ...##
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Page 2956 of Mathematical Reviews Vol. , Issue 81H
[page]

1981
*
Mathematical Reviews
*

sin:0 3069
The author gives a definition for a

*congruence*of a relational structure*and*then shows that the*congruences*are precisely all equivalence relations below a greatest*congruence*; an explicit ... Let now G be an infinite connected group of Morley rank n; if G is nilpotent, its class of nilpotency is at most n; if it is*solvable*its class is at most n*and*it has a normal series of*definable*subgroups ...##
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Pell's equations X2 − mY2 = −1, −4 and continued fractions

1986
*
Journal of Number Theory
*

Simple

doi:10.1016/0022-314x(86)90087-9
fatcat:crdbp7dvtbcf5lgphhlpxweiqy
*congruence*arguments show that, if (1.3) is*solvable*, then either m = 4 or 8 (mod $6) or ?n z 5 (mod 8)*and*that if (1.1) is*solvable*then m = 1 or 2 (mod 4). ... This*congruence*can be proved as follows: /(i(l + &)) = 1 (mod 2) e-xz-xwv--$(m- i)y'= -1 is*solvable*in integers x*and*y (see, e.g., [3, Sat2 3.351) e (2x-y)*-mSy2= -4 soIvable in integers .X*and*y -3 ...##
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On simultaneous representations of primes by binary quadratic forms

1984
*
Journal of Number Theory
*

Then p = M* + 7N2,

doi:10.1016/0022-314x(84)90111-2
fatcat:cdoggpj54jfmzditzdv3ybrqam
*and*knowing M*and*N makes it possible to "predict" whether p = A 2 + 14B2 is*solvable*or p = 7C* + 20' is*solvable*. ... Under appropriate assumptions, this information can be used to restrict the possible values of K for which K'p = A 2 + qrB2 is*solvable**and*the possible values of K' for which K12p = qCz + rD* is*solvable*... Shmuel Schreiber, who contributed numerous suggestions for greater clarity*and*accuracy. ...##
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Page 467 of Mathematical Reviews Vol. 18, Issue 6
[page]

1957
*
Mathematical Reviews
*

Then the

*congruence*b=a,x,*¥+ cee +a5x,* (mod pt+) is*solvable*for every b ¢ J[@]. Theorem 4. With the same notation as in the previous theorem*and*p|f, p? ... =1 which is cut off by the inequalities*defining*reduction. Let Xo be a simply connected region contained in Ao*and*having a piece-wise smooth boundary. ...##
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Page 32 of Mathematical Reviews Vol. 29, Issue 1
[page]

1965
*
Mathematical Reviews
*

Its

*principal**congruence*subgroup I'(n) consists of all MeT such that M=J,, (mod n). If a divides b,*define*the symplectic modulary group M(a, b)=I'(a)/T'(6). ... Now let A be an rx matrix, where r <n,*and**define*A to be primitive if its r x r minors generate the unit ideal of R. ...
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