Constructing Punctually Categorical Semigroups
- PDF / 138,900 Bytes
- 4 Pages / 594 x 792 pts Page_size
- 31 Downloads / 181 Views
Algebra and Logic, Vol. 59, No. 5, November, 2020 (Russian Original Vol. 59, No. 5, September-October, 2020)
CONSTRUCTING PUNCTUALLY CATEGORICAL SEMIGROUPS I. Sh. Kalimullin∗
UDC 512:7
Presented by the Program Committee of the Conference “Mal’tsev Readings” We construct a finitely generated semigroup that is primitive recursively isomorphic to any of its primitive recursive copies. According to [1], algebraic structures with such a property are said to be punctually categorical (see also [2, 3]). Definition. An algebraic structure in a finite signature is fully primitive recursive (or punctual) if all signature operations and predicates are primitive recursive, and the universe of the structure coincides with the set ω of all natural numbers. A punctual algebraic structure A is punctually categorical if for every punctual copy B ∼ =A −1 there exists a primitive recursive isomorphism g : A → B with a primitive recursive inverse g . Note that the above definition differs from the closely related concept of primitive recursive categoricity, which was dealt with, for instance, in [4-6], where copies of structures with a primitive recursive universe are taken into account. However, for structures that are not locally finite, the two approaches are equivalent [6]. In [2, 7], it was shown that it is rather difficult to find nontrivial examples of punctually categorical structures in natural algebraic classes. The structures in question do not exist in the classes of linear orders, Boolean algebras, Abelian p-groups, torsion-free Abelian groups, and fields. For finitely generated structures, a punctually categorical structure can be exemplified only by using two special unary operations on the structure. THEOREM 1 [2]. There exist a punctually categorical structure A in a functional signature {(a), s(x), t(x))} and a primitive recursive function f : ω → ω \ {0} for which the following statements hold: every element of A is uniquely representable as tm (sn (a)), where n ∈ ω and m < f (n); ∗
Supported by Russian Science Foundation, (project No. 18-11-00028) and by the Russian Ministry of Education and Science (project No. 1.451.2016/1.4).
Kazan (Volga Region) Federal University, Kazan, Russia; [email protected]. Translated from Algebra i Logika, Vol. 59, No. 5, pp. 600-605, September-October, 2020. Russian DOI: 10.33048/alglog.2020.59.507. Original article submitted February 11, 2020; accepted November 27, 2020.
408
c 2020 Springer Science+Business Media, LLC 0002-5232/20/5905-0408
tf (n) (sn (a)) = sn (a) for all n ∈ ω; s(tm (sn (a))) = sn+1 (a) for all n, m ∈ ω; moreover, f (2n) ≤ 2 for all n ∈ ω. We show that the given result can be applied for constructing a punctually categorical finitely generated semigroup. THEOREM 2. There exists a punctually categorical finitely generated semigroup. Of interest is the question whether there exists a punctually categorical group which differs from the trivial example—a direct sum of infinitely many copies of a fixed finite cyclic group of prime order. Also, the question remains open whethe
Data Loading...
