On the recursion theorem in iterative operative spaces

2001 ◽  
Vol 66 (4) ◽  
pp. 1727-1748 ◽  
Author(s):  
J. Zashev
Keyword(s):  
Large Class ◽  
Recursion Theory ◽  
Combinatory Logic ◽  
Fundamental Result ◽  
Intermediate Result ◽  
Full Generality ◽  
Partially Ordered ◽  
Ordered Algebras ◽  
Recursion Theorem

Abstract.The recursion theorem in abstract partially ordered algebras, such as operative spaces and others, is the most fundamental result of algebraic recursion theory. The primary aim of the present paper is to prove this theorem for iterative operative spaces in full generality. As an intermediate result, a new and rather large class of models of the combinatory logic is obtained.

scholarly journals The cohomology of semi-infinite Deligne–Lusztig varieties

2020 ◽  
Vol 2020 (768) ◽  
pp. 93-147
Author(s):  
Charlotte Chan
Keyword(s):  
Large Class ◽  
Finite Type ◽  
Cohomology Group ◽  
Geometric Realization ◽  
Local Fields ◽  
Irreducible Representations ◽  
Division Algebras ◽  
Full Generality ◽  
Homology Groups ◽  
Supercuspidal Representations

AbstractWe prove a 1979 conjecture of Lusztig on the cohomology of semi-infinite Deligne–Lusztig varieties attached to division algebras over local fields. We also prove the two conjectures of Boyarchenko on these varieties. It is known that in this setting, the semi-infinite Deligne–Lusztig varieties are ind-schemes comprised of limits of certain finite-type schemes {X_{h}}. Boyarchenko’s two conjectures are on the maximality of {X_{h}} and on the behavior of the torus-eigenspaces of their cohomology. Both of these conjectures were known in full generality only for division algebras with Hasse invariant {1/n} in the case {h=2} (the “lowest level”) by the work of Boyarchenko–Weinstein on the cohomology of a special affinoid in the Lubin–Tate tower. We prove that the number of rational points of {X_{h}} attains its Weil–Deligne bound, so that the cohomology of {X_{h}} is pure in a very strong sense. We prove that the torus-eigenspaces of the cohomology group {H_{c}^{i}(X_{h})} are irreducible representations and are supported in exactly one cohomological degree. Finally, we give a complete description of the homology groups of the semi-infinite Deligne–Lusztig varieties attached to any division algebra, thus giving a geometric realization of a large class of supercuspidal representations of these groups. Moreover, the correspondence {\theta\mapsto H_{c}^{i}(X_{h})[\theta]} agrees with local Langlands and Jacquet–Langlands correspondences. The techniques developed in this paper should be useful in studying these constructions for p-adic groups in general.

scholarly journals The spectral radius in partially ordered algebras

2006 ◽  
Vol 417 (2-3) ◽  
pp. 347-369 ◽  
Author(s):  
Thomas I. Seidman ◽  
Hans Schneider
Keyword(s):  
Spectral Radius ◽  
Partially Ordered ◽  
Ordered Algebras

Join-completions of partially ordered algebras

2020 ◽  
Vol 171 (10) ◽  
pp. 102842 ◽  
Author(s):  
José Gil-Férez ◽  
Luca Spada ◽  
Constantine Tsinakis ◽  
Hongjun Zhou
Keyword(s):  
Partially Ordered ◽  
Ordered Algebras

Quantum B-algebras

2013 ◽  
Vol 11 (11) ◽  
Author(s):  
Wolfgang Rump
Keyword(s):  
Number Theory ◽  
Algebraic Number Theory ◽  
Connected Components ◽  
Effect Algebras ◽  
Algebraic Semantics ◽  
Partially Ordered ◽  
Ordered Groups ◽  
Group A ◽  
Ordered Algebras ◽  
Mv Algebras

AbstractThe concept of quantale was created in 1984 to develop a framework for non-commutative spaces and quantum mechanics with a view toward non-commutative logic. The logic of quantales and its algebraic semantics manifests itself in a class of partially ordered algebras with a pair of implicational operations recently introduced as quantum B-algebras. Implicational algebras like pseudo-effect algebras, generalized BL- or MV-algebras, partially ordered groups, pseudo-BCK algebras, residuated posets, cone algebras, etc., are quantum B-algebras, and every quantum B-algebra can be recovered from its spectrum which is a quantale. By a two-fold application of the functor “spectrum”, it is shown that quantum B-algebras have a completion which is again a quantale. Every quantale Q is a quantum B-algebra, and its spectrum is a bigger quantale which repairs the deficiency of the inverse residuals of Q. The connected components of a quantum B-algebra are shown to be a group, a fact that applies to normal quantum B-algebras arising in algebraic number theory, as well as to pseudo-BCI algebras and quantum BL-algebras. The logic of quantum B-algebras is shown to be complete.

Recursion theory on orderings. II

1980 ◽  
Vol 45 (2) ◽  
pp. 317-333 ◽  
Author(s):  
J. B. Remmel
Keyword(s):  
Large Class ◽  
General Model ◽  
Recursion Theory ◽  
Second Order ◽  
Splitting Theorem ◽  
Natural Numbers ◽  
Partial Orderings

In [6], G. Metakides and the author introduced a general model theoretic setting in which to study the lattice of r.e. substructures of a large class of recursively presented models . Examples included , the natural numbers with equality, 〈 Q, ≤ 〉, the rationals under the usual ordering, and a large class of n-dimensional partial orderings. In this setting, we were able to generalize many of the constructions of classical recursion theory so that the constructions yield the classical results when we specialize to the case of and new results when we specialize to other models. Constructions to generalize Myhill's Theorem on creative sets [8], Friedberg's Theorem on the existence of maximal sets [3], Dekker's Theorem on the degrees of hypersimple sets [2], and Martin's Theorem on the degrees of maximal sets [5] were produced in [6]. In this paper, we give constructions to generalize the Morley-Soare Splitting Theorem [7] and Lachlan's characterization of hyperhypersimple sets [4] in §2, constructions to generalize Lachlan's theorems on the existence of major subsets and r-maximal sets contained in maximal sets [4] in §3, and constructions to generalize Robinson's construction of r-maximal sets that are not contained in any maximal sets [11] and second-order maximal sets [12] in §4.In §1 of this paper, we give the precise definitions of our model theoretic setting and deal with other preliminaries. Also in §1, we define the notions of “uniformly nonrecursive”, “uniformly maximal”, etc. which are the key notions involved in the generalizations of the various theorems that occur in §§2, 3 and 4.

scholarly journals Fixed points and self-reference

1984 ◽  
Vol 7 (2) ◽  
pp. 283-289 ◽  
Author(s):  
Raymond M. Smullyan
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Fixed Points ◽  
Point Theorem ◽  
Recursion Theory ◽  
Diagonal Argument ◽  
Recursion Theorem ◽  
Self Reference

It is shown how Gödel's famous diagonal argument and a generalization of the recursion theorem are derivable from a common construation. The abstract fixed point theorem of this article is independent of both metamathematics and recursion theory and is perfectly comprehensible to the non-specialist.

scholarly journals Relative Ockham lattices: their order-theoretic and algebraic characterisation

1990 ◽  
Vol 32 (1) ◽  
pp. 47-66 ◽  
Author(s):  
G. Bordalo ◽  
H. A. Priestley
Keyword(s):  
Distributive Lattice ◽  
Large Class ◽  
Closed Interval ◽  
Ordered Sets ◽  
Finite Algebras ◽  
Ordered Algebras

Given a variety of lattice-ordered algebras, a lattice L is said to be a relative-lattice if every closed interval [a, b] of L may be given the structure of an algebra in (in other words, is the reduct of a member of —not necessarily unique). This paper discusses the characterisation in terms of forbidden substructures of finite relative.stf-lattices. We treat a large class of varieties of distributive-lattice-ordered algebras. For these varieties, the finite algebras can be described dually in terms of finite ordered sets, so that order-theoretic results and techniques prove valuable.

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