Representations of Groups as Automorphisms on Orthomodular Lattices and Posets

Roughness in Orthomodular Lattices

International Journal of Mathematics and Soft Computing ◽

10.26708/ijmsc.2012.1.2.02 ◽

2012 ◽

Vol 2 (1) ◽

pp. 9

Author(s):

E. K. R Nagarajan ◽

D. Umadevi

Keyword(s):

Orthomodular Lattices

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Residuated Structures and Orthomodular Lattices

Studia Logica ◽

10.1007/s11225-021-09946-1 ◽

2021 ◽

Author(s):

D. Fazio ◽

A. Ledda ◽

F. Paoli

Keyword(s):

Algebraic Logic ◽

Residuated Lattices ◽

Quantum Structures ◽

Heyting Algebras ◽

Quantum Algebras ◽

Orthomodular Lattices ◽

Lattice Of Subvarieties ◽

Residuated Structures ◽

Common Framework ◽

Mv Algebras

AbstractThe variety of (pointed) residuated lattices includes a vast proportion of the classes of algebras that are relevant for algebraic logic, e.g., $$\ell $$ ℓ -groups, Heyting algebras, MV-algebras, or De Morgan monoids. Among the outliers, one counts orthomodular lattices and other varieties of quantum algebras. We suggest a common framework—pointed left-residuated $$\ell $$ ℓ -groupoids—where residuated structures and quantum structures can all be accommodated. We investigate the lattice of subvarieties of pointed left-residuated $$\ell $$ ℓ -groupoids, their ideals, and develop a theory of left nuclei. Finally, we extend some parts of the theory of join-completions of residuated $$\ell $$ ℓ -groupoids to the left-residuated case, giving a new proof of MacLaren’s theorem for orthomodular lattices.

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Rough Approximation Operators on a Complete Orthomodular Lattice

Axioms ◽

10.3390/axioms10030164 ◽

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.

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On finite degree partial representations of groups

Journal of Algebra ◽

10.1016/s0021-8693(03)00533-7 ◽

2004 ◽

Vol 274 (1) ◽

pp. 309-334 ◽

Cited By ~ 11

Author(s):

M. Dokuchaev ◽

N. Zhukavets

Keyword(s):

Finite Degree ◽

Representations Of Groups

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Logic in representations of groups

Algebra and Logic ◽

10.1007/s10469-012-9167-8 ◽

2012 ◽

Vol 51 (1) ◽

pp. 1-27

Author(s):

E. V. Aladova ◽

A. Gvaramiya ◽

B. Plotkin

Keyword(s):

Representations Of Groups

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VARIETIES OF REPRESENTATIONS OF GROUPS AND VARIETIES OF ASSOCIATIVE ALGEBRAS

International Journal of Algebra and Computation ◽

10.1142/s0218196711006881 ◽

2011 ◽

Vol 21 (07) ◽

pp. 1149-1178 ◽

Cited By ~ 1

Author(s):

ELENA ALADOVA ◽

BORIS PLOTKIN

Keyword(s):

General Theory ◽

Group Representations ◽

Associative Algebras ◽

Open Problems ◽

Representations Of Groups ◽

Main Emphasis

This paper is tightly connected with the book [Varieties of Group Representations. General Theory, Connections and Applications (Zinatne, Riga, 1983) (in Russian)]. In the paper we prove new results in the spirit of the above-mentioned book. They are related to dimension subgroups, varieties of representations of groups and varieties of associative algebras. The main emphasis is put on the varieties of representations of groups induced by the varieties of associative algebras. We provide the reader with the list of open problems. For many reasons we consciously included in the text a brief review of the basic definitions and results from the theory of varieties of representations described in the book [Varieties of Group Representations. General Theory, Connections and Applications].

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