PARTIAL GROUP ACTIONS ON SEMIALGEBRAS

For defining a K-semialgebra A, we use Katsov's tensor product which makes the category K-Smod monoidal. Further, if A is a K-semialgebra then AΔ is a KΔ-algebra and A embeds in AΔ. The subtractive and strong partial actions of a group are defined on A. A subtractive partial action α of a group G on A can be extended to a partial action of G on AΔ which helps in globalization of α. A strong partial action on A has a unique subtractive globalization. We also discuss the associativity of the skew group semiring A ×α G.

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SEMIALGEBRAS AND THEIR ALGEBRAS OF DIFFERENCES WITH PARTIAL GROUP ACTIONS ON THEM

Asian-European Journal of Mathematics ◽

10.1142/s1793557113500381 ◽

2013 ◽

Vol 06 (03) ◽

pp. 1350038

Author(s):

Ram Parkash Sharma ◽

Anu

Keyword(s):

Finite Group ◽

Prime Ideal ◽

Group Actions ◽

Rocky Mountain ◽

Prime Ideals ◽

Central Idempotent ◽

Group Rings ◽

Partial Action ◽

Partial Group ◽

A Finite Group

Let A be a semialgebra defined in [R. P. Sharma, Anu and N. Singh, Partial group actions on semialgebras, Asian European J. Math.5(4) (2012), Article ID:1250060, 20pp.] over an additively cancellative and commutative semiring K. In additively cancellative semirings, the subtractive ideals play an important role. If P is a subtractive and G-prime ideal of an additively cancellative and yoked semiring A, where G is a finite group acting on A, then A has finitely many n(≤ |G|) minimal primes over P (see [R. P. Sharma and T. R. Sharma, G-prime ideals in semirings and their skew group rings, Comm. Algebra34 (2006) 4459–4465], Lemma 3.6, a result analogous to Lemma 3.2 of Passman [D. S. Passman, It's essentially Maschke's theorem, Rocky Mountain J. Math.13 (1983) 37–54]). Consider a subtractive partial action α of a finite group G on A such that each Dg is generated by a central idempotent 1g of A and the intersection D = ⋂g∈G,Dg≠0Dg of nonzero Dg's is nonzero. It is not necessary that number of minimal primes in Spec A over a subtractive and α-prime ideal P of a yoked semiring A is less than or equal to the order of the group, if 1d ∈ P (Example 3.2). However, we show that the result is true if 1d ∉ P (Corollary 3.1). We also study the prime ideals of the partial fixed subsemiring Aα of A.

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GALOIS CORRESPONDENCES FOR PARTIAL GALOIS AZUMAYA EXTENSIONS

Journal of Algebra and Its Applications ◽

10.1142/s0219498811004999 ◽

2011 ◽

Vol 10 (05) ◽

pp. 835-847 ◽

Cited By ~ 8

Author(s):

ANTONIO PAQUES ◽

VIRGÍNIA RODRIGUES ◽

ALVERI SANT'ANA

Keyword(s):

Finite Group ◽

Galois Theory ◽

Group Actions ◽

Galois Extensions ◽

Partial Action ◽

Partial Group ◽

Azumaya Algebra ◽

Unital Ring ◽

One To One ◽

A Finite Group

Let α be a partial action, having globalization, of a finite group G on a unital ring R. Let Rα denote the subring of the α-invariant elements of R and CR(Rα) the centralizer of Rα in R. In this paper we will show that there are one-to-one correspondences among sets of suitable separable subalgebras of R, Rα and CR(Rα). In particular, we extend to the setting of partial group actions similar results due to DeMeyer [Some notes on the general Galois theory of rings, Osaka J. Math.2 (1965) 117–127], and Alfaro and Szeto [On Galois extensions of an Azumaya algebra, Commun. Algebra25 (1997) 1873–1882].

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THE EXISTENCE OF CONVEXITY IN PARTIAL GROUP ACTIONS

Universal Journal of Mathematics and Mathematical Sciences ◽

10.17654/um014010045 ◽

2021 ◽

Vol 14 (1) ◽

pp. 45-55

Author(s):

S. A. Adebisi ◽

M. Enioluwafe

Keyword(s):

Group Actions ◽

Partial Group

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Morita equivalence of partial group actions and globalization

Transactions of the American Mathematical Society ◽

10.1090/tran/6525 ◽

2015 ◽

Vol 368 (7) ◽

pp. 4957-4992 ◽

Cited By ~ 5

Author(s):

F. Abadie ◽

M. Dokuchaev ◽

R. Exel ◽

J. J. Simón

Keyword(s):

Group Actions ◽

Morita Equivalence ◽

Partial Group

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Inverse semigroup homomorphisms via partial group actions

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700019778 ◽

2001 ◽

Vol 64 (1) ◽

pp. 157-168 ◽

Cited By ~ 6

Author(s):

Benjamin Steinberg

Keyword(s):

Inverse Semigroup ◽

Group Actions ◽

Inverse Semigroups ◽

Inverse Monoid ◽

Partial Group ◽

Inverse Monoids ◽

Maximal Group ◽

Special Cases ◽

Quick Proof ◽

Group Component

This papar constructs all homomorphisms of inverse semigroups which factor through an E-unitary inverse semigroup; the construction is in terms of a semilattice component and a group component. It is shown that such homomorphisms have a unique factorisation βα with α preserving the maximal group image, β idempotent separating, and the domain I of β E-unitary; moreover, the P-representation of I is explicitly constructed. This theory, in particular, applies whenever the domain or codomain of a homomorphism is E-unitary. Stronger results are obtained for the case of F-inverse monoids.Special cases of our results include the P-theorem and the factorisation theorem for homomorphisms from E-unitary inverse semigroups (via idempotent pure followed by idempotent separating). We also deduce a criterion of McAlister–Reilly for the existence of E-unitary covers over a group, as well as a generalisation to F-inverse covers, allowing a quick proof that every inverse monoid has an F-inverse cover.

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Partial group actions on algebras

International Journal of Algebra ◽

10.12988/ija.2013.3555 ◽

2013 ◽

Vol 7 ◽

pp. 517-526

Author(s):

Ram Parkash Sharma ◽

Anu

Keyword(s):

Group Actions ◽

Partial Group

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PARTIAL ACTIONS OF GROUPS

International Journal of Algebra and Computation ◽

10.1142/s0218196704001657 ◽

2004 ◽

Vol 14 (01) ◽

pp. 87-114 ◽

Cited By ~ 51

Author(s):

J. KELLENDONK ◽

MARK V. LAWSON

Keyword(s):

Group Theory ◽

Inverse Semigroup ◽

Group Action ◽

Partial Function ◽

Inverse Semigroups ◽

Explicit Description ◽

Partial Action ◽

Partial Group

A partial action of a group G on a set X is a weakening of the usual notion of a group action: the function G×X→X that defines a group action is replaced by a partial function; in addition, the existence of g·(h·x) implies the existence of (gh)·x, but not necessarily conversely. Such partial actions are extremely widespread in mathematics, and the main aim of this paper is to prove two basic results concerning them. First, we obtain an explicit description of Exel's universal inverse semigroup [Formula: see text], which has the property that partial actions of the group G give rise to actions of the inverse semigroup [Formula: see text]. We apply this result to the theory of graph immersions. Second, we prove that each partial group action is the restriction of a universal global group action. We describe some applications of this result to group theory and the theory of E-unitary inverse semigroups.

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Anosov automorphisms for nilmanifolds and rigidity of group actions

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700008415 ◽

1995 ◽

Vol 15 (2) ◽

pp. 341-359 ◽

Cited By ~ 2

Author(s):

Nantian Qian

Keyword(s):

Tensor Product ◽

Lyapunov Exponents ◽

Finite Index ◽

Group Actions ◽

Irreducible Representations ◽

Product Representation ◽

Tensor Product Representation

AbstractWe obtain the density of Lyapunov exponents for maximal abelian ℝ-split group in kth tensor product representation of a subgroup Γ ⊂ SL(n, ℤ) of finite index under certain conditions. Anosov and Cartan actions of such groups associated with irreducible representations of SL(n, ℝ) are also classified. Examples of rigidity of actions on nilmanifolds are discussed.

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