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Definable principal congruences and solvability

Paweł M. Idziak, Keith A. Kearnes, Emil W. Kiss, Matthew A. Valeriote
2009 Annals of Pure and Applied Logic  
We prove that in a locally finite variety that has 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.  ... 
doi:10.1016/j.apal.2008.09.019 fatcat:6tbmha4mevdtbdqb74kd2hlnga

Inherently nonfinitely based solvable algebras

Keith Kearnes, Ross Willard
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 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.  ... 
doi:10.4153/cmb-1994-074-6 fatcat:662ytnrhirgz7dfch2426chgzi

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,  ... 

Ray class field extensions of real quadratic fields and solvability of congruences

Paul B. Massell
1985 Journal of Number Theory  
An explicit description is given of a process using classical class field theory to generate 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.  ... 
doi:10.1016/0022-314x(85)90021-6 fatcat:grktksux2rgpjecaldwzatty2a

Commutator theory for racks and quandles [article]

Marco Bonatto, David Stanovský
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  ... 
arXiv:1902.08980v3 fatcat:hug723twdfd3fhgvnhldfqsq6y

Residual finiteness growths of Lamplighter groups [article]

Khalid Bou-Rabee and Junjie Chen and Anastasiia Timashova
2019 arXiv   pre-print
In our proof, we quantify a 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.  ... 
arXiv:1909.03535v2 fatcat:lcy3pjohpzhxfnimlfxrhbrjgi

The commutator in equivalential algebras and Fregean varieties

Paweł M. Idziak, Katarzyna Słomczyńska, Andrzej Wroński
2011 Algebra Universalis  
In this paper we give a full characterization of the commutator for equivalential algebras 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.  ... 
doi:10.1007/s00012-011-0133-4 fatcat:uchepwrlm5gyxjnjp7mecolrqq

Mal'cev classes of left-quasigroups and Quandles [article]

Marco Bonatto, Stefano Fioravanti
2021 arXiv   pre-print
According to the commutator theory developed in [FM87] 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  ... 
arXiv:2004.05368v2 fatcat:c6iomhvhangghpzo3yng4gvtiu

Congruences in Algebraic Number Fields Involving Sums of Similar Powers

Eckford Cohen
1956 Transactions of the American Mathematical Society  
In the final section ( §5) estimates for Q,(p) are obtained (Theorems 6 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).  ... 
doi:10.2307/1992888 fatcat:qx3doactrjcmte55mihii4sopa

Congruences in algebraic number fields involving sums of similar powers

Eckford Cohen
1956 Transactions of the American Mathematical Society  
In the final section ( §5) estimates for Q,(p) are obtained (Theorems 6 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).  ... 
doi:10.1090/s0002-9947-1956-0084524-3 fatcat:ynkf7tnoujan7eb2g6dnhvfpxe

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  ... 

Pell's equations X2 − mY2 = −1, −4 and continued fractions

Pierre Kaplan, Kenneth S Williams
1986 Journal of Number Theory  
Simple 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  ... 
doi:10.1016/0022-314x(86)90087-9 fatcat:crdbp7dvtbcf5lgphhlpxweiqy

On simultaneous representations of primes by binary quadratic forms

Joseph B. Muskat
1984 Journal of Number Theory  
Then p = M* + 7N2, 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.  ... 
doi:10.1016/0022-314x(84)90111-2 fatcat:cdoggpj54jfmzditzdv3ybrqam

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.  ... 

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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