A Finiteness Criterion for Orthomodular Lattices

1978 ◽  
Vol 30 (02) ◽  
pp. 315-320 ◽  
Author(s):  
Günter Bruns
Keyword(s):  
Boolean Algebra ◽  
Orthomodular Lattice ◽  
Infinite Chain ◽  
The Other ◽  
Finitely Generated ◽  
Orthomodular Lattices ◽  
Other Hand ◽  
Finiteness Criterion ◽  
Element Set

The main result of this paper is the following: THEOREM. Every finitely generated orthomodular lattice L with finitely many maximal Boolean subalgebras (blocks) is finite. If L has one block only, our theorem reduces to the well-known fact that every finitely generated Boolean algebra is finite. On the other hand, it is known that a finitely generated orthomodular lattice without any further restrictions can be infinite. In fact, in [2] we constructed an orthomodular lattice which is generated by a three-element set with two comparable elements, has infinitely many blocks and contains an infinite chain.

scholarly journals Rough Approximation Operators on a Complete Orthomodular Lattice

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 164
Author(s):  
Songsong Dai
Keyword(s):  
Boolean Algebra ◽  
Orthomodular Lattice ◽  
Orthomodular Lattices ◽  
Approximation Operators ◽  
Rough Approximation ◽  
Distributive Law ◽  
Complete Orthomodular Lattice ◽  
The Relationship

This paper studies rough approximation via join and meet on a complete orthomodular lattice. Different from Boolean algebra, the distributive law of join over meet does not hold in orthomodular lattices. Some properties of rough approximation rely on the distributive law. Furthermore, we study the relationship among the distributive law, rough approximation and orthomodular lattice-valued relation.

scholarly journals Generating Adjoint Groups

2019 ◽  
Vol 62 (3) ◽  
pp. 733-738 ◽  
Author(s):  
Be'eri Greenfeld
Keyword(s):  
Open Problem ◽  
Jacobson Radical ◽  
The Other ◽  
Adjoint Group ◽  
Finitely Generated ◽  
Infinite Dimensional ◽  
Other Hand ◽  
Nil Ring

AbstractWe prove two approximations of the open problem of whether the adjoint group of a non-nilpotent nil ring can be finitely generated. We show that the adjoint group of a non-nilpotent Jacobson radical cannot be boundedly generated and, on the other hand, construct a finitely generated, infinite-dimensional nil algebra whose adjoint group is generated by elements of bounded torsion.

Power-collapsing games

2008 ◽  
Vol 73 (4) ◽  
pp. 1433-1457 ◽  
Author(s):  
Miloš S. Kurilić ◽  
Boris Šobot
Keyword(s):  
Boolean Algebra ◽  
Dense Subset ◽  
Winning Strategy ◽  
Zero Element ◽  
The Other ◽  
Complete Boolean Algebra ◽  
Infinite Cardinal ◽  
Other Hand ◽  
Game Theoretic

AbstractThe game is played on a complete Boolean algebra , by two players. White and Black, in κ-many moves (where κ is an infinite cardinal). At the beginning White chooses a non-zero element p ∈ . In the α-th move White chooses pα ∈ (0, p) and Black responds choosing iα ∈{0, 1}. White winsthe play iff . where and .The corresponding game theoretic properties of c.B.a.'s are investigated. So, Black has a winning strategy (w.s.) if κ ≥ π() or if contains a κ-closed dense subset. On the other hand, if White has a w.s., then κ ∈ . The existence of w.s. is characterized in a combinatorial way and in terms of forcing. In particular, if 2<κ = κ ∈ Reg and forcing by preserves the regularity of κ, then White has a w.s. iff the power 2κ is collapsed to κ in some extension. It is shown that, under the GCH, for each set S ⊆ Reg there is a c.B.a. such that White (respectively. Black) has a w.s. for each infinite cardinal κ ∈ S (resp. κ ∉ S). Also it is shown consistent that for each κ ∈ Reg there is a c.B.a. on which the game is undetermined.

scholarly journals Four games on Boolean algebras

Filomat ◽  
2016 ◽  
Vol 30 (13) ◽  
pp. 3389-3395
Author(s):  
Milos Kurilic ◽  
Boris Sobot
Keyword(s):  
Boolean Algebra ◽  
Winning Strategy ◽  
Zero Element ◽  
Boolean Algebras ◽  
The Other ◽  
Complete Boolean Algebra ◽  
Other Hand

The games G2 and G3 are played on a complete Boolean algebra B in ?-many moves. At the beginning White picks a non-zero element p of B and, in the n-th move, White picks a positive pn < p and Black chooses an in ? {0,1}. White wins G2 iff lim inf pin,n = 0 and wins G3 iff W A?[?]? ? n?A pin,n = 0. It is shown that White has a winning strategy in the game G2 iff White has a winning strategy in the cut-and-choose game Gc&c introduced by Jech. Also, White has a winning strategy in the game G3 iff forcing by B produces a subset R of the tree <?2 containing either ??0 or ??1, for each ? ? <?2, and having unsupported intersection with each branch of the tree <?2 belonging to V. On the other hand, if forcing by B produces independent (splitting) reals then White has a winning strategy in the game G3 played on B. It is shown that ? implies the existence of an algebra on which these games are undetermined.

scholarly journals Centralising Monoids with Low-Arity Witnesses on a Four-Element Set

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1471
Author(s):  
Mike Behrisch ◽  
Edith Vargas-García
Keyword(s):  
The Other ◽  
Full Transformation ◽  
Group Operation ◽  
Binary Operations ◽  
Other Hand ◽  
Transformation Monoid ◽  
Element Set

As part of a project to identify all maximal centralising monoids on a four-element set, we determine all centralising monoids witnessed by unary or by idempotent binary operations on a four-element set. Moreover, we show that every centralising monoid on a set with at least four elements witnessed by the Maľcev operation of a Boolean group operation is always a maximal centralising monoid, i.e., a co-atom below the full transformation monoid. On the other hand, we also prove that centralising monoids witnessed by certain types of permutations or retractive operations can never be maximal.

Stability and posets

2009 ◽  
Vol 74 (2) ◽  
pp. 693-711 ◽  
Author(s):  
Carl G. Jockusch ◽  
Bart Kastermans ◽  
Steffen Lempp ◽  
Manuel Lerman ◽  
Reed Solomon
Keyword(s):  
Reverse Mathematics ◽  
Infinite Chain ◽  
The Other ◽  
Weak Stability ◽  
Other Hand ◽  
Natural Notion ◽  
Weakly Stable

AbstractHirschfeldt and Shore have introduced a notion of stability for infinite posets. We define an arguably more natural notion called weak stability, and we study the existence of infinite computable or low chains or antichains, and of infinite chains and antichains, in infinite computable stable and weakly stable posets. For example, we extend a result of Hirschfeldt and Shore to show that every infinite computable weakly stable poset contains either an infinite low chain or an infinite computable antichain. Our hardest result is that there is an infinite computable weakly stable poset with no infinite chains or antichains. On the other hand, it is easily seen that every infinite computable stable poset contains an infinite computable chain or an infinite antichain. In Reverse Mathematics, we show that SCAC, the principle that every infinite stable poset contains an infinite chain or antichain, is equivalent over RCA0 to WSAC, the corresponding principle for weakly stable posets.

States and Boolean Algebra

2008 ◽  
Vol 15 (04) ◽  
pp. 649-652
Author(s):  
Nabila N. Mikhaeel ◽  
Basim Samir Labib
Keyword(s):  
Mathematical Logic ◽  
Boolean Algebra ◽  
Orthomodular Lattice ◽  
Orthomodular Lattices ◽  
Quantum Theories

We investigate subadditive measures on orthomodular lattices. We show as the main result that the Boolean algebra, the special metric orthomodular lattice and the orthomodular lattice which is unital with respect to subadditive states are equivalent. This result may find an application in the foundation of quantum theories and mathematical logic.

scholarly journals The Complexity of Classification Problems for Models of Arithmetic

2010 ◽  
Vol 16 (3) ◽  
pp. 345-358 ◽  
Author(s):  
Samuel Coskey ◽  
Roman Kossak
Keyword(s):  
Classification Problem ◽  
The Other ◽  
Saturated Model ◽  
Classification Problems ◽  
Finitely Generated ◽  
Saturated Models ◽  
Other Hand ◽  
Recursively Saturated ◽  
Recursively Saturated Model ◽  
Models Of Arithmetic

AbstractWe observe that the classification problem for countable models of arithmetic is Borel complete. On the other hand, the classification problems for finitely generated models of arithmetic and for recursively saturated models of arithmetic are Borel; we investigate the precise complexity of each of these. Finally, we show that the classification problem for pairs of recursively saturated models and for automorphisms of a fixed recursively saturated model are Borel complete.

scholarly journals Maximal perpendicularity in certain Abelian groups

2017 ◽  
Vol 9 (1) ◽  
pp. 235-247
Author(s):  
Mika Mattila ◽  
Jorma K. Merikoski ◽  
Pentti Haukkanen ◽  
Timo Tossavainen
Keyword(s):  
Abelian Group ◽  
Vector Space ◽  
Binary Relation ◽  
Abelian Groups ◽  
The Other ◽  
Finitely Generated ◽  
Other Hand

AbstractWe define perpendicularity in an Abelian group G as a binary relation satisfying certain five axioms. Such a relation is maximal if it is not a subrelation of any other perpendicularity in G. A motivation for the study is that the poset (𝒫, ⊆) of all perpendicularities in G is a lattice if G has a unique maximal perpendicularity, and only a meet-semilattice if not. We study the cardinality of the set of maximal perpendicularities and, on the other hand, conditions on the existence of a unique maximal perpendicularity in the following cases: G ≅ ℤn, G is finite, G is finitely generated, and G = ℤ ⊕ ℤ ⊕ ⋯. A few such conditions are found and a few conjectured. In studying ℝn, we encounter perpendicularity in a vector space.

THE NUMBER OF PROFINITE GROUPS WITH A SPECIFIED SYLOW SUBGROUP

2015 ◽  
Vol 99 (1) ◽  
pp. 108-127
Author(s):  
COLIN D. REID
Keyword(s):  
Normal Subgroup ◽  
Sylow Subgroup ◽  
The Other ◽  
Profinite Groups ◽  
Finitely Generated ◽  
Soluble Groups ◽  
Other Hand ◽  
Formula Group ◽  
Nontrivial Normal Subgroup

Let $S$ be a finitely generated pro-$p$ group. Let ${\mathcal{E}}_{p^{\prime }}(S)$ be the class of profinite groups $G$ that have $S$ as a Sylow subgroup, and such that $S$ intersects nontrivially with every nontrivial normal subgroup of $G$. In this paper, we investigate whether or not there is a bound on $|G:S|$ for $G\in {\mathcal{E}}_{p^{\prime }}(S)$. For instance, we give an example where ${\mathcal{E}}_{p^{\prime }}(S)$ contains an infinite ascending chain of soluble groups, and on the other hand show that $|G:S|$ is bounded in the case where $S$ is just infinite.

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