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Clausal relations and {C}-clones

Edith Vargas
2010 Discussiones Mathematicae - General Algebra and Applications  
power of D, i.e.  ...  In 2.2.8 we saw that clausal relations are either trivial or non-trivial, and none of the diagonals except for the full Cartesian powers are clausal relations.  ...  Thus, Pol R Due to 6.1.5 and 3.4.4, we get Note that one non-trivial clausal relation of arity two can be used to describe both a co-atom in the lattice of C -clones and a co-atom in the lattice of C -  ... 
doi:10.7151/dmgaa.1167 fatcat:6lclwqalkvfn7dkwol3ca2ed5m

On polymorphism-homogeneous relational structures and their clones

Christian Pech, Maja Pech
2014 Algebra Universalis  
We study polymorphism-homogeneity from a model-theoretic, an algebraic, and a combinatorial point of view.  ...  It turns out that this notion (under various names and in completely different contexts) has been existing in algebraic literature for at least 30 years.  ...  Let A be a countable set and let L be a meet-complete sublattice of E(A). Then L is polymorphism-homogeneous if and only if it is arithmetical.  ... 
doi:10.1007/s00012-014-0310-3 fatcat:mx6av4chwjgupaoxxum5wzdqri

No-Cloning in Reduced Power Algebras [article]

Elemer E Rosinger
2009 arXiv   pre-print
many reduced power algebras which can replace the usual real numbers R, or complex numbers C.  ...  The No-Cloning property in Quantum Computation is known not to depend on the unitarity of the operators involved, but only on their linearity.  ...  wide and appropriate framework of scalars, namely, any one of the infinitely many algebras which belong to the class of reduced power algebras, [4] [5] [6] [7] [8] [9] [10] [11] [12] .  ... 
arXiv:0902.0264v1 fatcat:6th5bqouvraefiqugmntc6jlca

Special Relativity in Reduced Power Algebras [article]

Elemer E Rosinger
2010 arXiv   pre-print
Recently, [10,11], the Heisenberg Uncertainty relation and the No-Cloning property in Quantum Mechanics and Quantum Computation, respectively, have been extended to versions of Quantum Mechanics and Quantum  ...  Suffice to mention two of them : the difficult problem of so called "infinities in Physics" falls easily aside due to the presence of infinitesimal and infinitely large scalars in such reduced power algebras  ...  Increased and Decreased Precision in Measurements As a general issue, relating not only to Relativity or the Quanta, the presence of infinitesimal and infinitely large scalars in reduced power algebras  ... 
arXiv:0903.0296v2 fatcat:nkyufrngujfahcpqynhnjgf7ia

Reflections and powers of multisorted minions

Erkko Lehtonen, Reinhard Pöschel
2021 Algebra Universalis  
AbstractClasses of multisorted minions closed under extensions, reflections, and direct powers are considered from a relational point of view.  ...  As a generalization of a result of Barto, Opršal, and Pinsker, the closure of a multisorted minion is characterized in terms of constructions on multisorted relation pairs which are invariant for minions  ...  The well-known Galois connection Pol-Inv (for clones and relational clones, induced by the property of preservation of a relation by a function) does not work well for describing these minions as Galois  ... 
doi:10.1007/s00012-020-00707-0 fatcat:eoe5uzusdbdptaevrkugo2awwa

Bounding the free spectrum of nilpotent algebras of prime power order

Erhard Aichinger
2019 Israel Journal of Mathematics  
If A is of prime power order, then it is known that there is a polynomial p such that for every n ∈N, every n-generated algebra in the variety generated by A has at most 2^p(n) elements.  ...  Let A be a finite nilpotent algebra in a congruence modular variety with finitely many fundamental operations.  ...  Now assume that A is of prime power order. Then G is of prime power order and squarefree exponent, and therefore elementary abelian.  ... 
doi:10.1007/s11856-019-1846-x fatcat:ijcj7vexx5drdfbdsvhgniefoe

Regular elements and green's relations in power Menger algebras of terms

K. Denecke, P. Glubudom
2008 Demonstratio Mathematica  
Idempotent and regular elements with respect to the diagonal power semigroup There is another algebra associated with the Menger algebra (P(W τ (X n ));Ŝ n ), the diagonal power semigroup (P(W τ (X n )  ...  The algebra P n − clone τ := (P(W τ (X n ));Ŝ n , {x 1 }, . . . , {x n }) is a unitary Menger algebra of rank n (see e.g.  ... 
doi:10.1515/dema-2013-0055 fatcat:ho2ex4qnwjhzdjq546lsc7wmay

On the number of countable subdirect powers of unary algebras [article]

Nik Ruskuc, Bill de Witt
2021 arXiv   pre-print
A finite unary algebra (A,F) has only countably many countable subdirect powers if and only if every operation f∈ F is either a permutation or a constant mapping.  ...  That A is subdirectly irreducible follows from the fact that it has a single proper congruence, namely the equivalence relation with classes {0, 1} and {2}, using the standard criterion for subidirect  ...  The clone itself consists of compositions of the elements of Mon(A) and projections.  ... 
arXiv:2106.08876v1 fatcat:4oh3ew5ahbbzpa3f4y6y5at6bm

Representing varieties of algebras by algebras

Walter D. Neumann
1970 Journal of the Australian Mathematical Society  
The ])$ 0 -clone M of the variety W of monoids is generated by two elements m and e satisfying relations (3), (5), (6) , and (8) .  ...  Neumann [8] One can show that 2l tnj is the vaiiety of algebras whose underlying sets can be written canonicaily as a power A n of an Si-algebra A, and whose homomorphisms are the maps of the form A  ... 
doi:10.1017/s1446788700005899 fatcat:yazuqxsiv5fdffwxap2jscl46a

REGULAR ELEMENTS AND GREEN'S RELATIONS IN POWER MENGER ALGEBRAS OF TERMS

K. Denecke, P. Glubudom
2008 Demonstratio Mathematica  
For the corresponding semigroups of tree languages we study idempotent and regular elements as well as Green's relations C and 1Z. Sm : WT(Xn) X (WT(Xm)) n -WT(Xm).  ...  On sets of tree languages superposition operations can be defined in such a way that the collection of all tree languages of type r forms an abstract clone.  ...  [1] ) that this algebra satisfies the clone axioms There is a homogeneous analogue of this structure, formed as follows.  ... 
doi:10.1515/dema-2008-0103 fatcat:l4usd3ylcnbqloi5ayzde75fme

Expressive Power, Satisfiability and Equivalence of Circuits over Nilpotent Algebras

Pawel M. Idziak, Piotr Kawalek, Jacek Krzaczkowski, Michael Wagner
2018 International Symposium on Mathematical Foundations of Computer Science  
This in turn is one of the oldest and well-known mathematical problems which for centuries was the driving force of research in algebra.  ...  We go outside Boolean realm and consider circuits built of any fixed set of gates on an arbitrary large finite domain.  ...  A is k-nilpotent, decomposes into a direct product of algebras of prime power order and the clone Clo A is generated by finitely many operations, 3.  ... 
doi:10.4230/lipics.mfcs.2018.17 dblp:conf/mfcs/IdziakKK18 fatcat:tlfybhasbzberbhmby5kdiw7we

Structure identification of Boolean relations and plain bases for co-clones

Nadia Creignou, Phokion Kolaitis, Bruno Zanuttini
2008 Journal of computer and system sciences (Print)  
Our algorithm is derived by first introducing the concept of a plain basis for a co-clone and then identifying natural plain bases for every co-clone in Post's lattice.  ...  We give a quadratic algorithm for the following structure identification problem: given a Boolean relation R and a finite set S of Boolean relations, can the relation R be expressed as a conjunctive query  ...  In universal algebra, ∃CNF(S)-formulas are known as primitive positive formulas, and they play an important role in the Galois connection between clones of functions and co-clones of relations [15, 16  ... 
doi:10.1016/j.jcss.2008.02.005 fatcat:byizubbfjff7fc5yartl7wx5ha

Universal Clone Algebra [article]

Antonino Salibra
2022 arXiv   pre-print
We provide concrete examples showing that general results in Universal Clone Algebra, when translated in terms of algebras and clones, give more powerful versions of known theorems in Universal Algebra  ...  Algebras and clones of finitary operations are to Universal Algebra what t-algebras and clone algebras are to Universal Clone Algebra.  ...  Salibra Functional clone algebras and t-algebras The following proposition describes the existing relations between a t-algebra A and the functional clone algebras of value domain A.  ... 
arXiv:2203.14054v2 fatcat:vlmtbbax45cipchpur2eh6me2u

The Convolution Algebra [article]

John Harding, Carol Walker, Elbert Walker
2017 arXiv   pre-print
For a complete lattice L and a relational structure X=(X,(R_i)_I), we introduce the convolution algebra L^X.  ...  This algebra consists of the lattice L^X equipped with an additional n_i-ary operation f_i for each n_i+1-ary relation R_i of X. For α_1,...,α_n_i∈ L^X and x∈ X we set f_i(α_1,...  ...  Let X = (X, (R i ) I ) be a relational structure of type τ , and let P(X) be the power set of X.  ... 
arXiv:1702.02847v1 fatcat:ncz32ta4jbaxxjbddffzwkdq6q

Algebras from Congruences [article]

Peter Mayr, Agnes Szendrei
2021 arXiv   pre-print
to the interval of congruences between 0 and α on A, this isomorphism preserves higher commutators and TCT types, and C inherits all idempotent Maltsev conditions from A.  ...  We present a functorial construction which, starting from a congruence α of finite index in an algebra A, yields a new algebra C with the following properties: the congruence lattice of C is isomorphic  ...  following relational clone operations: intersection of relations of the same arity, composition, and coordinate manipulations (permutation, identification, and duplication of coordinates, and projection  ... 
arXiv:1910.00689v2 fatcat:ep3tqiuvpvcoriuj256ppthxke
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