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An algebraic approach to unification under associativity and commutativity

1987
*
Journal of symbolic computation
*

From the work of Siekmann & Livesey,

doi:10.1016/s0747-7171(87)80001-9
fatcat:amr4ubrmxfgc5f2q3vbzf3aiqq
*and*Stickel it is known how*to*unify two terms in*an**associative**and**commutative*theory: transfer the terms into Abelian strings, look for mappings which solve the ... The following work formalises the transformations between the free*algebra**and*this monoid. This leads*to**an*algorithm which uses maximal information for its search for solutions in the monoid. ... One possible solution*to*this problem is*to*build*associativity**and**commutativity*into the matching*and**unification*algorithms. ...##
###
Adding homomorphisms to commutative/monoidal theories or how algebra can help in equational unification
[chapter]

1991
*
Lecture Notes in Computer Science
*

With this paper we pursue the semantic

doi:10.1007/3-540-53904-2_91
fatcat:6jzlkhtkxnccfmbmwzr3xccnra
*approach**to**unification*. ... The two results illustrate how using*algebraic*machinery can lead*to*general results*and*elegant proofs in*unification*theory. ... We are now ready*to*give*an**algebraic*definition of*commutative*theories. ...##
###
Overview of the structural unification of quantum mechanics and relativity using the algebra of quantions
[article]

2012
*
arXiv
*
pre-print

Lack of division makes possible structural

arXiv:0901.0332v3
fatcat:zfhtf3ljcrdbxkykco5jnrrdqe
*unification*with relativity (one cannot upgrade the linear Minkowski space*to*a division*algebra*due*to*null light-cone vectors)*and*demands*an*adjustment from ... The purpose of this contribution is*to*provide*an*introduction for a general physics audience*to*the recent results of Emile Grgin that unifies quantum mechanics*and*relativity into the same mathematical ... This remains*an*important open problem*to*be solved*under*a complete unified theory. ...##
###
Nonassociative gauge fields
[article]

2020
*
arXiv
*
pre-print

It turns out that such

arXiv:2003.10709v2
fatcat:inbnmnoolrdixjhypnisbyqhky
*an**approach*allows us*to*build the gauge-Higgs*unification*group with the*unification*scale equal*to*electroweak*and*obtain the value of the Weinberg angle in very good agreement ... In this paper, we study consequences of the assumption that the gauge group SU(2) of the standard model is a non-*associative*image of Spin(3). ... It is easy*to*obtain these representations if we consider A G as a subalgebra of the*commutator**algebra*of H. Let the representation of G be*associative*. ...##
###
Page 3261 of Mathematical Reviews Vol. , Issue 87f
[page]

1987
*
Mathematical Reviews
*

This

*approach*is simpler*and*more efficient than the*approaches*based on generalizations of the Knuth- Bendix completion proce- dure*to*handle*associative**and**commutative*operators. ... Summary: “We develop a new*approach*based on computing the Grobner basis of polynomial ideals for solving word problems*and**unification*problems for finitely presented*commutative**algebras*. ...##
###
Unification theory

1989
*
Journal of symbolic computation
*

This article surveys what is presently known in

doi:10.1016/s0747-7171(89)80012-4
fatcat:mjwsvawtirdvbc5dxdflp4n2zi
*unification*theory*and*records its early history. 20g ... This operation, called*unification*in work on deduction, is the "addition-*and*-multiplication" of AI-systems*and*is consequently often supported by special purpose hardware or by a fast instruction set ... But*and*is*an**associative**and**commutative*operation*and*the natural join obeys*an**associative*axiom, which distributes over some other operation, hence both can be built into a*unification*algorithm ( ...##
###
A Tour through Non-Associative Geometry
[article]

1997
*
arXiv
*
pre-print

Moreover, we comment on the application of our method

arXiv:hep-th/9607086v2
fatcat:xihsahgxzjgg7gbdjlt76u7evq
*to*the standard model*and*the flipped SU(5) x U(1)-grand*unification*model. ... This concept shares common features with the non-*commutative*geometry a la Connes/Lott, differs from that, however, by the implementation of skew-adjoint Lie*algebras*instead of unital*associative**-*algebras*... The perhaps most successful NCG-*approach*towards grand*unification*was proposed by Chamseddine, Felder*and*Fröhlich. In the SU(5)-model [4, 5] , the authors start*to*construct*an*auxiliary K-cycle. ...##
###
Lectures on Unification
[article]

1997
*
arXiv
*
pre-print

In

arXiv:hep-ph/9712206v1
fatcat:xzqouxv42baitmlpm6oa3wbav4
*an*appendix, we review the group theory pertinent*to*the program. ... In these lectures we review the motivation, principles of*and*(circumstantial) evidence for the program of*unification*of the fundamental forces. ... In particular, modern*approaches**to**unification*simultaneously require grand*unification*as well as*unification*of bosonic*and*fermionic statistics*and*is called supersymmetric*unification**and*is the framework ...##
###
On the noncommutative standard model
[article]

2007
*
arXiv
*
pre-print

*An*almost

*commutative*geometry can be viewed as

*an*ordinary (

*commutative*) 4-dimensional space-time with

*an*'internal' Kaluza-Klein space attached

*to*each point. ... It is now possible

*to*relax the

*commutativity*of the

*algebra*A. ...

##
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How to prevent type-flaw attacks on security protocols under algebraic properties
[article]

2010
*
arXiv
*
pre-print

We also discuss how tagging could be used

arXiv:1003.5385v1
fatcat:5let6o24lbhizlzazoxbnw2goe
*to*prevent type-flaw attacks*under*other properties such as*associativity*of pairing,*commutative*encryption, prefix property*and*homomorphic encryption. ... Heather et al. proved that tagging every message field with it's type prevents all type-flaw attacks*under*a free message*algebra**and*perfect encryption system. ... We benefited greatly from the following people's help*and*...##
###
Page 2413 of Mathematical Reviews Vol. , Issue 96d
[page]

1996
*
Mathematical Reviews
*

for function symbols which are

*associative**and**commutative*, or*associative*, 68Q Theory of computing 964:68114*commutative**and*idempotent.” ... Yuji Kobayashi (J-TOHOS-IS; Funabashi) 96d:68113 68Q42 03B15 03B40 03C05 Dougherty, Daniel J. (1-WESL; Middletown, CT); Johann, Patricia A combinatory logic*approach**to*higher-order E -*unification*. ...##
###
PSEUDO-LINEAR ALGEBRA: A POWERFUL TOOL IN UNIFICATION OF THE STUDY OF NONLINEAR CONTROL SYSTEMS

2007
*
IFAC Proceedings Volumes
*

This paper introduces methods of pseudo-linear

doi:10.3182/20070822-3-za-2920.00117
fatcat:v537o6ziynaj7oosjke45wymr4
*algebra**to*unify the*algebraic*formalism of one-forms*and*the related polynomial*approach*for both continuous*and*discrete-time nonlinear control systems. ... Given*approach*covers also differene, q-shift*and*q-difference operators whereby this*algebraic*formalism is not only unified but also extended*to*wide class of nonlinear control systems. ...*Under*(7) , σ is*an*automorphism of K*and*there exists, up*to**an*isomorphism, a unique difference field K * called the inversive closure of K (Cohn, 1965) . ...##
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Noncommutative geometry, Grand Symmetry and twisted spectral triple

2015
*
Journal of Physics, Conference Series
*

In the noncommutative geometry

doi:10.1088/1742-6596/634/1/012008
fatcat:kqywinrtzvfdnjbkby7kgjvjga
*approach**to*the standard model we discuss the possibility*to*derive the extra scalar field sv- initially suggested by particle physicist*to*stabilize the electroweak vacuum ... - from a "grand*algebra*" that contains the usual standard model*algebra*. ... Acknowledgments The author thanks W. van Suijlekom*and*F. Lizzi for discussions. ...##
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Syntactic Unification as a Geometric Operation in Free Modules over certain Rings

2002
*
Electronical Notes in Theoretical Computer Science
*

We have shown elsewhere how

doi:10.1016/s1571-0661(04)80512-7
fatcat:nmegsbibnreazokqitnrxfrgk4
*to*introduce a concept of syntactic*unification*when terms are taken as the elements in a free module*and*established the link between both*unification*concepts showing that ... ,*under*certain reasonable hypotheses, they are completely equivalent. ... This ring may be shown*to*be a DF-ring (see [5] )*and*thus appropriate*to*build*an*implementation of a term*algebra*on a free module over it. ...##
###
Page 4984 of Mathematical Reviews Vol. , Issue 86k
[page]

1986
*
Mathematical Reviews
*

(Ordering corresponds

*to*successive approximations of a computation.) The*algebraic*semantics*approach*allows*unification*of syntax*and*semantics. ... H.(1-IASU) 86k:08001 * Modal theory:*an**algebraic**approach**to*order, geometry,*and*convexity. R & E Research*and*Exposition in Mathematics, 9. Heldermann Verlag, Berlin, 1985. xii+158 pp. DM 38.00. ...
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