A note on the multiplicative semigroup of models of Peano arithmetic

1989 ◽  
Vol 54 (3) ◽  
pp. 936-940
Author(s):  
Roman Kossak ◽  
Mark Nadel ◽  
James Schmerl
Keyword(s):  
Additive Group ◽  
Peano Arithmetic ◽  
Strong Result ◽  
Isomorphism Type ◽  
Multiplicative Semigroup ◽  
Additive Semigroup ◽  
History Of

In a model of Peano arithmetic, the isomorphism type of the multiplicative semigroup uniquely determines the isomorphism type of the additive semigroup. In fact, for any prime p of , the function x ↦ px is an isomorphism of the additive semigroup with the multiplicative subsemigroup of powers of p. It was observed by Jensen and Ehrenfeucht [3] that for countable models of PA, the isomorphism type of the additive semigroup (or even the additive group) determines the isomorphism type of the multiplicative semigroup. (See Theorem 3 below.)In this note we will show that the countability restriction cannot be dropped. First, we show (as Theorem 2) that for uncountable models of PA the isomorphism type of the additive group never determines the isomorphism type of the multiplicative semigroup. Our main result is Theorem 5 in which we show that the isomorphism type of the additive semigroup need not determine the isomorphism type of the multiplicative semigroup, thereby improving upon Harnik [2], where Theorem 5 is proved under the assumption of ♢. For completeness, a sketch of the proof of the Jensen-Ehrenfeucht result is included.The history of this paper begins with Nadel's question, asked in 1981, whether the countability assumption can be eliminated in the Jensen-Ehrenfeucht theorem. Soon afterwards, Nadel obtained the strong counterexample of Theorem 2, which applied to the additive group rather than the additive semigroup. A result of Pabion [8] shows that such a strong result is not possible for the additive semigroup.

scholarly journals Proof of Fermat's Last Theorem By Infinite Descent

2020 ◽  
Author(s):  
Djamel Himane
Keyword(s):  
20Th Century ◽  
Elementary Function ◽  
Mathematical Problem ◽  
History Of Mathematics ◽  
Peano Arithmetic ◽  
Basic Principles ◽  
Fermat's Last Theorem ◽  
Fermat’S Last Theorem ◽  
History Of ◽  
Infinite Descent

Fermat's last theorem, one of the most challenging theories in the history of mathematics, has been conjectured by French lawyer Pierre de Verma in 1637. Since then, it wasconsidered the most difficult and unsolvable mathematical problem. However, more than three centuries later, a first proof was proposed by the British mathematician Andrew Wiles in 1994, relying on 20th-century techniques. Wiles's proof is based on elliptic (oval) curves that were not available at the time when the theory was first proposed. Most mathematicians argued that it was impossible to prove Fermat's theorem according to basic principles of arithmetic, though Harvey Friedman's grand conjecture states that mathematical theorems, including Fermat's Last Theorem, can be solved in very weak systems such as the Elementary Function Arithmetic (EFA). Friedman's grand conjecture states that "every theorem published in the journal, Annals of Mathematics, whose statement involves only finitary mathematical objects (i.e., what logicians call an arithmetical statement) can be proved in EFA, which is the weak fragment of Peano Arithmetic based on the usual quantifier free axioms for 0,1,+,x, exp, together with thescheme of induction for all formulas in the language all of whose quantifiers are bounded." *

scholarly journals Automorphism groups of laminated near-rings

1980 ◽  
Vol 23 (1) ◽  
pp. 97-102 ◽  
Author(s):  
K. D. Magill
Keyword(s):  
Automorphism Groups ◽  
Additive Group ◽  
Multiplicative Semigroup ◽  
Natural Way

Let N be an arbitrary near-ring. Each element a ∈ N determines in a natural way a new multiplication on the elements of N which results in a near-ring Na whose additive group coincides with that of N but whose multiplicative semigroup generally differs. Specifically, we define the product x * y of two elements in Na by x * y = x a y where a product in the original near-ring is denoted by juxtaposition. One easily checks that Na is a near-ring with addition identical to that of N. The original near-ring N will be referred to as the base near-ring, Na will be referred to as a laminated near-ring of N and a will be referred to as the laminating element or sometimes more simply as the laminator.

scholarly journals The semigroup of nonempty finite subsets of rationals

1988 ◽  
Vol 11 (1) ◽  
pp. 81-86
Author(s):  
Reuben Spake
Keyword(s):  
Direct Product ◽  
Additive Group ◽  
Rational Numbers ◽  
Additive Semigroup ◽  
Semilattice Decomposition ◽  
Archimedean Component

LetQbe the additive group of rational numbers and letℛbe the additive semigroup of all nonempty finite subsets ofQ. ForX∈ℛ, defineAXto be the basis of〈X−min(X)〉andBXthe basis of〈max(X)−X〉. In the greatest semilattice decomposition ofℛ, let𝒜(X)denote the archimedean component containingX. In this paper we examine the structure ofℛand determine its greatest semilattice decomposition. In particular, we show that forX,Y∈ℛ,𝒜(X)=𝒜(Y)if and only ifAX=AYandBX=BY. Furthermore, ifXis a non-singleton, then the idempotent-free𝒜(X)is isomorphic to the direct product of a power joined subsemigroup and the groupQ.

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