EQUATIONAL THEORIES GENERATED BY HYPERSUBSTITUTIONS OF TYPE (n)

2002 ◽  
Vol 12 (06) ◽  
pp. 867-876 ◽  
Author(s):  
K. DENECKE ◽  
J. KOPPITZ ◽  
ST. NIWCZYK
Keyword(s):  
Free Term ◽  
Operation Symbol ◽  
Arbitrary Type ◽  
Equational Theories ◽  
Invariant Congruence ◽  
Congruence Relations ◽  
Term Algebra ◽  
Fully Invariant Congruence

Hypersubstitutions map n-ary operation symbols to n-ary terms. Such mappings can be uniquely extended to mappings defined on the set of all terms. It turns out that the kernels of hypersubstitutions are fully invariant congruence relations on the (absolutely free) term algebra of the considered type. For an arbitrary type τ = (n), n ≥ 1, i.e. if one has only one n-ary operation symbol, we will describe all these congruence relations. The results will be applied to solve the hyperunification problem. Further we will give some generalizations to arbitrary types.

scholarly journals On equivalence of semigroup identities

2001 ◽  
Vol 88 (2) ◽  
pp. 161 ◽  
Author(s):  
O. Macedońska ◽  
M. Żabka
Keyword(s):  
Free Semigroup ◽  
Positive Answer ◽  
The Other ◽  
Invariant Congruence ◽  
Semigroup Identities ◽  
Fully Invariant Congruence

For a given relation $\rho$ on a free semigroup ${\mathcal F}$ we describe the smallest cancellative fully invariant congruence ${\rho}^{\sharp}$ containing $\rho$. Two semigroup identities are s-equivalent if each of them is a consequence of the other on cancellative semigroups. If two semigroup identities are equivalent on groups, it is not known if they are s-equivalent. We give a positive answer to this question for all binary semigroup identities of the degree less or equal to 5. A poset of corresponding varieties of groups is given.

Derived Varieties and Derived Equational Theories

1998 ◽  
Vol 08 (02) ◽  
pp. 153-169 ◽  
Author(s):  
K. Denecke ◽  
J. Koppitz ◽  
R. Marszałek
Keyword(s):  
Final Section ◽  
Operation Symbol ◽  
Equational Theories ◽  
Solid Variety ◽  
Derived Algebras ◽  
Derivation Process

This paper describes a derivation process for varieties and equational theories using the theory of hypersubstitutions and M-hyperidentities. A hypersubstitution σ of type τ is a map which takes each n-ary operation symbol of the type to an n-ary term of this type. If [Formula: see text] is an algebra of type τ then the algebra [Formula: see text] is called a derived algebra of [Formula: see text]. If V is a class of algebras of type τ then one can consider the variety vσ(V) generated by the class of all derived algebras from V. In the first two sections the necessary definitions are given. In Sec. 3 the properties of derived varieties and derived equational theories are described. On the set of all derived varieties of a given variety, a quasiorder is developed which gives a derivation diagram. In the final section the derivation diagram for the largest solid variety of medial semigroups is worked out.

GENERALIZED DERIVED ALGEBRAS AND GENERALIZED INDUCED ALGEBRAS

2010 ◽  
Vol 03 (03) ◽  
pp. 485-494
Author(s):  
Sarawut Phuapong ◽  
Sorasak Leeratanavalee
Keyword(s):  
Congruence Relation ◽  
Equational Theory ◽  
Invariant Congruence ◽  
Fully Invariant Congruence ◽  
Derived Algebras

Substituting for the fundamental operations of an algebra term operations we get a new algebra of the same type, called a generalized derived algebra. Such substitutions are called generalized hypersubstitutions. Generalized hypersubstitutions can also be applied to every equation of a fully invariant equational theory. The equational theory generated by the resulting set of the equations induces on every algebra of the type under consideration a fully invariant congruence relation. If we factorize the generalized derived algebra by this fully invariant congruence relation we will obtain an algebra which we call generalized induced algebra. In this paper, we prove some properties which transfer the starting algebras to generalized derived algebras and to generalized induced algebras.

scholarly journals Ramanujan’s function k(τ)=r(τ)r 2(2τ) and its modularity

2020 ◽  
Vol 18 (1) ◽  
pp. 1727-1741
Author(s):  
Yoonjin Lee ◽  
Yoon Kyung Park
Keyword(s):  
Continued Fraction ◽  
Quadratic Field ◽  
Singular Values ◽  
Modular Function ◽  
Imaginary Quadratic Field ◽  
Modular Equations ◽  
Positive Rational Number ◽  
Congruence Relations ◽  
Symmetry Relations ◽  
Ray Class Field

Abstract We study the modularity of Ramanujan’s function k ( τ ) = r ( τ ) r 2 ( 2 τ ) k(\tau )=r(\tau ){r}^{2}(2\tau ) , where r ( τ ) r(\tau ) is the Rogers-Ramanujan continued fraction. We first find the modular equation of k ( τ ) k(\tau ) of “an” level, and we obtain some symmetry relations and some congruence relations which are satisfied by the modular equations; these relations are quite useful for reduction of the computation cost for finding the modular equations. We also show that for some τ \tau in an imaginary quadratic field, the value k ( τ ) k(\tau ) generates the ray class field over an imaginary quadratic field modulo 10; this is because the function k is a generator of the field of the modular function on Γ 1 ( 10 ) {{\mathrm{\Gamma}}}_{1}(10) . Furthermore, we suggest a rather optimal way of evaluating the singular values of k ( τ ) k(\tau ) using the modular equations in the following two ways: one is that if j ( τ ) j(\tau ) is the elliptic modular function, then one can explicitly evaluate the value k ( τ ) k(\tau ) , and the other is that once the value k ( τ ) k(\tau ) is given, we can obtain the value k ( r τ ) k(r\tau ) for any positive rational number r immediately.

scholarly journals The Limits of Computation

Axiomathes ◽  
2021 ◽  
Author(s):  
Andrew Powell
Keyword(s):  
Natural Number ◽  
Set Theory ◽  
Lambda Calculus ◽  
Second Order ◽  
Functional Interpretation ◽  
Arbitrary Type ◽  
Computable Functions

AbstractThis article provides a survey of key papers that characterise computable functions, but also provides some novel insights as follows. It is argued that the power of algorithms is at least as strong as functions that can be proved to be totally computable in type-theoretic translations of subsystems of second-order Zermelo Fraenkel set theory. Moreover, it is claimed that typed systems of the lambda calculus give rise naturally to a functional interpretation of rich systems of types and to a hierarchy of ordinal recursive functionals of arbitrary type that can be reduced by substitution to natural number functions.

Congruence relations on almost distributive lattices-II

2019 ◽  
pp. 2150005
Author(s):  
Gezahagne Mulat Addis
Keyword(s):  
Distributive Lattice ◽  
Congruence Relation ◽  
Congruence Class ◽  
Distributive Lattices ◽  
Ideal Formula ◽  
Congruence Relations

For a given ideal [Formula: see text] of an almost distributive lattice [Formula: see text], we study the smallest and the largest congruence relation on [Formula: see text] having [Formula: see text] as a congruence class.

Theories with equational forking

2002 ◽  
Vol 67 (1) ◽  
pp. 326-340 ◽  
Author(s):  
Markus Junker ◽  
Ingo Kraus
Keyword(s):  
Symmetric Relation ◽  
Equational Theories

AbstractWe show that equational independence in the sense of Srour equals local non-forking. We then examine so-called almost equational theories where equational independence is a symmetric relation.

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