scholarly journals Canonical forms for free κ-semigroups

2014 ◽  
Vol Vol. 16 no. 1 (Automata, Logic and Semantics) ◽  
Author(s):  
José Carlos Costa
Keyword(s):  
Normal Form ◽  
Word Problem ◽  
Canonical Form ◽  
Finite Semigroup ◽  
Finite Alphabet ◽  
Canonical Forms ◽  
Finite Semigroups ◽  
International Audience

Automata, Logic and Semantics International audience The implicit signature κ consists of the multiplication and the (ω-1)-power. We describe a procedure to transform each κ-term over a finite alphabet A into a certain canonical form and show that different canonical forms have different interpretations over some finite semigroup. The procedure of construction of the canonical forms, which is inspired in McCammond\textquoterights normal form algorithm for ω-terms interpreted over the pseudovariety A of all finite aperiodic semigroups, consists in applying elementary changes determined by an elementary set Σ of pseudoidentities. As an application, we deduce that the variety of κ-semigroups generated by the pseudovariety S of all finite semigroups is defined by the set Σ and that the free κ-semigroup generated by the alphabet A in that variety has decidable word problem. Furthermore, we show that each ω-term has a unique ω-term in canonical form with the same value over A. In particular, the canonical forms provide new, simpler, representatives for ω-terms interpreted over that pseudovariety.

Undecidability of the identity problem for finite semigroups

1992 ◽  
Vol 57 (1) ◽  
pp. 179-192 ◽  
Author(s):  
Douglas Albert ◽  
Robert Baldinger ◽  
John Rhodes
Keyword(s):  
Word Problem ◽  
Finite Semigroup ◽  
Identity Problem ◽  
Finite Semigroups ◽  
Or Groups ◽  
Finite Set ◽  
Variety Of Semigroups ◽  
Finite Products

In 1947 E. Post [28] and A. A. Markov [18] independently proved the undecidability of the word problem (or the problem of deducibility of relations) for semigroups. In 1968 V. L. Murskil [23] proved the undecidability of the identity problem (or the problem of deducibility of identities) in semigroups.If we slightly generalize the statement of these results we can state many related results in the literature and state our new results proved here. Let V denote either a (Birkhoff) variety of semigroups or groups or a pseudovariety of finite semigroups. By a very well-known theorem a (Birkhoff) variety is defined by equations or equivalently closed under substructure, surmorphisms and all products; see [7]. It is also well known that V is a pseudovariety of finite semigroups iff V is closed under substructure, surmorphism and finite products, or, equivalently, determined eventually by equations w1 = w1′, w2 = w2′, w3 = w3′,… (where the finite semigroup S eventually satisfies these equations iff there exists an n, depending on S, such that S satisfies Wj = Wj′ for j ≥ n). See [8] and [29]. All semigroups form a variety while all finite semigroups form a pseudovariety.We now consider a table (see the next page). In it, for example, the box denoting the “word” (identity) problem for the psuedovariety V” means, given a finite set of relations (identities) E and a relation (identity) u = ν, the problem of whether it is decidable that E implies u = ν inside V.

Single premise Post canonical forms defined over one-letter alphabets

1974 ◽  
Vol 39 (3) ◽  
pp. 489-495 ◽  
Author(s):  
Charles E. Hughes
Keyword(s):  
Word Problem ◽  
Canonical Form ◽  
Decision Problem ◽  
Decision Problems ◽  
Canonical Forms ◽  
Restricted Class ◽  
Constructive Proofs

AbstractIn this paper we investigate some families of decision problems associated with a restricted class of Post canonical forms, specifically, those defined over one-letter alphabets whose productions have single premises and contain only one variable. For brevity sake, we call any such form an RPCF (Restricted Post Canonical Form). Constructive proofs are given which show, for any prescribed nonrecursive r.e. many-one degree of unsolvability D, the existence of an RPCF whose word problem is of degree D and an RPCF with axiom whose-decision problem is also of degree D. Finally, we show that both of these results are best possible in that they do not hold for one-one degrees.

scholarly journals A proof of Zhil'tsov's theorem on decidability of equational theory of epigroups

2016 ◽  
Vol Vol. 17 no. 3 (Combinatorics) ◽  
Author(s):  
Inna Mikhaylova
Keyword(s):  
Finite Semigroup ◽  
Equational Theory ◽  
Unary Operation ◽  
Finitely Based ◽  
Infinite Basis ◽  
International Audience

International audience Epigroups are semigroups equipped with an additional unary operation called pseudoinversion. Each finite semigroup can be considered as an epigroup. We prove the following theorem announced by Zhil'tsov in 2000: the equational theory of the class of all epigroups coincides with the equational theory of the class of all finite epigroups and is decidable. We show that the theory is not finitely based but provide a transparent infinite basis for it.

scholarly journals Reduced Canonical Forms of Stoppers

10.37236/1083 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Aaron N. Siegel
Keyword(s):  
Canonical Form ◽  
Canonical Forms ◽  
Alternate Proof

The reduced canonical form of a loopfree game $G$ is the simplest game infinitesimally close to $G$. Reduced canonical forms were introduced by Calistrate, and Grossman and Siegel provided an alternate proof of their existence. In this paper, we show that the Grossman–Siegel construction generalizes to find reduced canonical forms of certain loopy games.

Canonical forms for definable subsets of algebraically closed and real closed valued fields

1995 ◽  
Vol 60 (3) ◽  
pp. 843-860 ◽  
Author(s):  
Jan E. Holly
Keyword(s):  
Canonical Form ◽  
Simple Form ◽  
Finite Union ◽  
Canonical Forms ◽  
Boolean Combination ◽  
Valued Fields ◽  
The Real

AbstractWe present a canonical form for definable subsets of algebraically closed valued fields by means of decompositions into sets of a simple form, and do the same for definable subsets of real closed valued fields. Both cases involve discs, forming “Swiss cheeses” in the algebraically closed case, and cuts in the real closed case. As a step in the development, we give a proof for the fact that in “most” valued fields F, if f(x), g(x) ∈ F[x] and v is the valuation map, then the set {x: v(f(x)) ≤ v(g(x))} is a Boolean combination of discs; in fact, it is a finite union of Swiss cheeses. The development also depends on the introduction of “valued trees”, which we define formally.

VI.—On Singular Pencils of Zehfuss, Compound, and Schläflian Matrices

1937 ◽  
Vol 56 ◽  
pp. 50-89 ◽  
Author(s):  
W. Ledermann
Keyword(s):  
Canonical Form ◽  
Direct Product ◽  
Canonical Forms ◽  
Singular Case ◽  
Elementary Divisors ◽  
Compound Matrices ◽  
Matrix Pencils ◽  
The Subject

In this paper the canonical form of matrix pencils will be discussed which are based on a pair of direct product matrices (Zehfuss matrices), compound matrices, or Schläflian matrices derived from given pencils whose canonical forms are known.When all pencils concerned are non-singular (i.e. when their determinants do not vanish identically), the problem is equivalent to finding the elementary divisors of the pencil. This has been solved by Aitken (1935), Littlewood (1935), and Roth (1934). In the singular case, however, the so-called minimal indices or Kronecker Invariants have to be determined in addition to the elementary divisors (Turnbull and Aitken, 1932, chap. ix). The solution of this problem is the subject of the following investigation.

scholarly journals Integrability analysis of the partial differential equation describing the classical bond-pricing model of mathematical finance

Open Physics ◽  
2018 ◽  
Vol 16 (1) ◽  
pp. 766-779
Author(s):  
Taha Aziz ◽  
Aeeman Fatima ◽  
Chaudry Masood Khalique
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Heat Equation ◽  
Canonical Form ◽  
Mathematical Finance ◽  
Fundamental Solutions ◽  
Bond Pricing ◽  
Canonical Forms ◽  
Pricing Model ◽  
Invariant Approach

AbstractThe invariant approach is employed to solve the Cauchy problem for the bond-pricing partial differential equation (PDE) of mathematical finance. We first briefly review the invariant criteria for a scalar second-order parabolic PDE in two independent variables and then utilize it to reduce the bond-pricing equation to different Lie canonical forms. We show that the invariant approach aids in transforming the bond-pricing equation to the second Lie canonical form and that with a proper parametric selection, the bond-pricing PDE can be converted to the first Lie canonical form which is the classical heat equation. Different cases are deduced for which the original equation reduces to the first and second Lie canonical forms. For each of the cases, we work out the transformations which map the bond-pricing equation into the heat equation and also to the second Lie canonical form. We construct the fundamental solutions for the bond-pricing model via these transformations by utilizing the fundamental solutions of the classical heat equation as well as solution to the second Lie canonical form. Finally, the closed-form analytical solutions of the Cauchy initial value problems for the bond-pricing model with proper choice of terminal conditions are obtained.

scholarly journals Growth sequences of finite semigroups

1987 ◽  
Vol 43 (1) ◽  
pp. 16-20 ◽  
Author(s):  
James Wiegold ◽  
H. Lausch
Keyword(s):  
Real Number ◽  
Finite Semigroup ◽  
Piecewise Linear ◽  
The Other ◽  
Direct Power ◽  
Growth Sequence ◽  
Group Of Units ◽  
Number Of Generators ◽  
Finite Semigroups ◽  
Minimum Number

AbstractThe growth sequence of a finite semigroup S is the sequence {d(Sn)}, where Sn is the nth direct power of S and d stands for minimum generating number. When S has an identity, d(Sn) = d(Tn) + kn for all n, where T is the group of units and k is the minimum number of generators of S mod T. Thus d(Sn) is essentially known since d(Tn) is (see reference 4), and indeed d(Sn) is then eventually piecewise linear. On the other hand, if S has no identity, there exists a real number c > 1 such that d(Sn) ≥ cn for all n ≥ 2.

Potential Divisibility in Finite Semigroups is Undecidable

1998 ◽  
Vol 08 (06) ◽  
pp. 671-679 ◽  
Author(s):  
Stanislav Kublanovsky ◽  
Mark Sapir
Keyword(s):  
Finite Semigroup ◽  
Finite Semigroups

We prove that there is no algorithm to decide, given a finite semigroup S and two elements a, b∈S, whether there exists a bigger finite semigroup T>S where a divides b and b divides a. This solves a thirty years old problem by John Rhodes.

EVANS' NORMAL FORM THEOREM REVISITED

2007 ◽  
Vol 17 (08) ◽  
pp. 1577-1592 ◽  
Author(s):  
JONATHAN D. H. SMITH
Keyword(s):  
Normal Form ◽  
Symmetric Group ◽  
Word Problem ◽  
Latin Squares ◽  
Form Theorem ◽  
Left Regular ◽  
Normal Form Theorem ◽  
Partial Latin Squares

Evans defined quasigroups equationally, and proved a Normal Form Theorem solving the word problem for free extensions of partial Latin squares. In this paper, quasigroups are redefined as algebras with six basic operations related by triality, manifested as coupled right and left regular actions of the symmetric group on three symbols. Triality leads to considerable simplifications in the proof of Evans' Normal Form Theorem, and makes it directly applicable to each of the six major varieties of quasigroups defined by subgroups of the symmetric group. Normal form theorems for the corresponding varieties of idempotent quasigroups are obtained as immediate corollaries.

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