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An algebraic approach to unification under associativity and commutativity

Albrecht Fortenbacher
1987 Journal of symbolic computation  
From the work of Siekmann & Livesey, 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.  ... 
doi:10.1016/s0747-7171(87)80001-9 fatcat:amr4ubrmxfgc5f2q3vbzf3aiqq

Adding homomorphisms to commutative/monoidal theories or how algebra can help in equational unification [chapter]

Franz Baader, Werner Nutt
1991 Lecture Notes in Computer Science  
With this paper we pursue the semantic 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.  ... 
doi:10.1007/3-540-53904-2_91 fatcat:6jzlkhtkxnccfmbmwzr3xccnra

Overview of the structural unification of quantum mechanics and relativity using the algebra of quantions [article]

Florin Moldoveanu
2012 arXiv   pre-print
Lack of division makes possible structural 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.  ... 
arXiv:0901.0332v3 fatcat:zfhtf3ljcrdbxkykco5jnrrdqe

Nonassociative gauge fields [article]

E.K. Loginov
2020 arXiv   pre-print
It turns out that such 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.  ... 
arXiv:2003.10709v2 fatcat:inbnmnoolrdixjhypnisbyqhky

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

Jörg H. Siekmann
1989 Journal of symbolic computation  
This article surveys what is presently known in 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 (  ... 
doi:10.1016/s0747-7171(89)80012-4 fatcat:mjwsvawtirdvbc5dxdflp4n2zi

A Tour through Non-Associative Geometry [article]

Raimar Wulkenhaar
1997 arXiv   pre-print
Moreover, we comment on the application of our method 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.  ... 
arXiv:hep-th/9607086v2 fatcat:xihsahgxzjgg7gbdjlt76u7evq

Lectures on Unification [article]

B. Ananthanarayan (Centre for Theoretical Studies, Indian Institute of Science, Bangalore, India)
1997 arXiv   pre-print
In 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  ... 
arXiv:hep-ph/9712206v1 fatcat:xzqouxv42baitmlpm6oa3wbav4

On the noncommutative standard model [article]

Jan-H. Jureit, Thomas Krajewski, Thomas Schucker, Christoph A. Stephan
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.  ... 
arXiv:0705.0489v1 fatcat:l2newckiijbuxgvjjwmppsfsne

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

Sreekanth Malladi, Pascal Lafourcade
2010 arXiv   pre-print
We also discuss how tagging could be used 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  ... 
arXiv:1003.5385v1 fatcat:5let6o24lbhizlzazoxbnw2goe

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

Miroslav Halás, Ülle Kotta
2007 IFAC Proceedings Volumes  
This paper introduces methods of pseudo-linear 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) .  ... 
doi:10.3182/20070822-3-za-2920.00117 fatcat:v537o6ziynaj7oosjke45wymr4

Noncommutative geometry, Grand Symmetry and twisted spectral triple

Agostino Devastato
2015 Journal of Physics, Conference Series  
In the noncommutative geometry 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.  ... 
doi:10.1088/1742-6596/634/1/012008 fatcat:kqywinrtzvfdnjbkby7kgjvjga

Syntactic Unification as a Geometric Operation in Free Modules over certain Rings

R.N. Banerjee, A. Bujosa
2002 Electronical Notes in Theoretical Computer Science  
We have shown elsewhere how 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.  ... 
doi:10.1016/s1571-0661(04)80512-7 fatcat:nmegsbibnreazokqitnrxfrgk4

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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