Rogers Semilattices of Families of Arithmetic Sets (original) (raw)
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Algebraic Properties of Rogers Semilattices of Arithmetical Numberings
Computability and Models, 2003
We investigate initial segments and intervals of Rogers semilattices of arithmetical families. We prove that there exist intervals with different algebraic properties; the elementary theory of any Rogers semilattice at arithmetical level n ≥ 2 is hereditarily undecidable; the class of all Rogers semilattices of a fixed level n ≥ 2 has an incomplete theory. * All authors were partially supported by grant INTAS-RFBR Computability and Models no. 97-139 and grant INTAS Computability in Hierarchies and Topological Spaces no. 00-499.
ELEMENTARY PROPERTIES OF ROGERS SEMILATTICES OF ARITHMETICAL NUMBERINGS
2004
We investigate differences in the elementary theories of Rogers semilattices of arithmetical numberings, depending on structural invariants of the given families of arithmetical sets. It is shown that at any fixed level of the arithmetical hierarchy there exist infinitely many families with pairwise elementary different Rogers semilattices.
Elementary Theories for Rogers Semilattices
Algebra and Logic, 2005
It is proved that for every level of the arithmetic hierarchy, there exist infinitely many families of sets with pairwise non-elementarily equivalent Rogers semilattices.
Isomorphism types and theories of Rogers semilattices of arithmetical numberings
2003
We investigate differences in isomorphism types and elementary theories of Rogers semilattices of arithmetical numberings, depending on different levels of the arithmetical hierarchy. It is proved that new types of isomorphism appear as the arithmetical level increases. It is also proved the incompleteness of the theory of the class of all Rogers semilattices of any fixed level. Finally, no Rogers semilattice of any infinite family at arithmetical level n ≥ 2 is weakly distributive, whereas Rogers semilattices of finite families are always distributive.
Some Characterizations of 0-DISTRIBUTIVE Semilattices
2012
In this paper we discuss prime down-sets of a semilattice. We give a characterization of prime down-sets of a semilattice. We also give some characterizations of 0-distributive semilattices and a characterization of minimal prime ideals containing an ideal of a 0-distributive semilattice. Finally, we give a characterization of minimal prime ideals of a pseudocomplemented semilattice.
Canadian Journal of Mathematics, 1980
Throughout this paper the ring R and the semigroup S are commutative with identity; moreover, it is assumed that S is cancellative, i.e., that S can be embedded in a group. The aim of this note is to determine necessary and sufficient conditions on R and 5 that the semigroup ring R[S] should be one of the following types of rings: principal ideal ring (PIR), ZPI-ring, Bezout, semihereditary or arithmetical. These results shed some light on the structure of semigroup rings and provide a source of examples of the rings listed above. They also play a key role in the determination of all commutative reduced arithmetical semigroup rings (without the cancellative hypothesis on S) which will appear in a forthcoming paper by Leo Chouinard and the authors [4]. Our results are motivated in large part by the paper [11] of R. Gilmer and T. Parker. In particular, Theorem 1.1 of [11] asserts that if R and S are as above and, moreover, if 5 is torsion-free, then the following are equivalent conditions: (1) R[S] is a Bezout ring; (2) R[S] is a Priifer ring; (3) R is a (von Neumann) regular ring and 5 is isomorphic to either a subgroup of the additive rationals or the positive cone of such a subgroup. One could very naturally include a fourth condition, namely: (4) R[S] is arithmetical. L. Fuchs [7] defines an arithmetical ring as a commutative ring with identity for which the ideals form a distributive lattice. Since a Priifer ring is one for which (A + B) C\ C = {A C\ C) + (B Pi C) whenever at least one of the ideals A, B or C contains a regular element (see [18]), arithmetical rings are certainly Priifer. On the other hand, it is well known that every Bezout ring is arithmetical, so that (4) is indeed equivalent to (l)-(3) in Theorem 1.1. In Theorem 3.6 of this paper we drop the requirement that S be torsion-free and determine necessary and sufficient conditions for the semigroup ring of a cancellative semigroup to be arithmetical. Examples are included to show that for these more general semigroup rings, the equivalences of the torsion-free case are no longer true. Theorems 4.1 and 4.2 provide characterizations of semigroup rings that are ZPI-rings and PIR's. Again, the corresponding results in [18] for torsion-free semigroups fail to hold in the more general case. We would like to thank Leo Chouinard for showing us how to remove
A decomposition of the Rogers semilattice of a family of dce sets
Journal of Symbolic Logic, 2009
Khutoretskii's Theorem states that the Rogers semilattice of any family of ce sets has either at most one or infinitely many elements. A lemma in the inductive step of the proof shows that no Rogers semilattice can be partitioned into a principal ideal and a principal ...