Department of Mathematics
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Item Affine Near-Semirings Over Brandt Semigroups(Taylor & Francis, 2014) Kumar, JitenderIn order to study the structure of A +(B n )—the affine near-semiring over a Brandt semigroup—this work completely characterizes the Green's classes of its semigroup reducts. In this connection, this work classifies the elements of A +(B n ) and reports the size of A +(B n ). Further, idempotents and regular elements of the semigroup reducts of A +(B n ) have also been characterized and studied some relevant semigroups in A +(B n ).Item The large rank of a finite semigroup using prime subsets(Springer, 2014-03) Kumar, JitenderThe large rank of a finite semigroup , denoted by r5( ), is the least number n such that every subset of with n elements generates . Howie and Ribeiro showed that r5( ) = |V| + 1, where V is a largest proper subsemigroup of . This work considers the complementary concept of subsemigroups, called prime subsets, and gives an alternative approach to find the large rank of a finite semigroup. In this connection, the paper provides a shorter proof of Howie and Ribeiro’s result about the large rank of Brandt semigroups. Further, this work obtains the large rank of the semigroup of order-preserving singular selfmapsItem Radicals and Ideals of Affine Near-semirings over Brandt Semigroups(ARXIV, 2015-06) Kumar, JitenderThis work obtains all the right ideals, radicals, congruences and ideals of the affine near-semirings over Brandt semigroups.Item Syntactic semigroup problem for the semigroup reducts of affine near-semirings over Brandt semigroups(World Scientific, 2016) Kumar, JitenderThe syntactic semigroup problem is to decide whether a given finite semigroup is syntactic or not. This work investigates the syntactic semigroup problem for both the semigroup reducts of A+(Bn), the affine near-semiring over a Brandt semigroup Bn. It is ascertained that both the semigroup reducts of A+(Bn) are syntactic semigroups.Item Rank properties of the semigroup of endomorphisms over Brandt semigroup(Springer, 2017-10) Kumar, JitenderSince the work of Marczewski [10], many authors have studied the rank properties in the context of general algebras (cf. [1–3,9,11,12,14]). The concept of rank for general algebras is analogous to the concept of dimension in linear algebra. The dimension of a vector space is the maximum cardinality of an independent subset, or equivalently, it is the minimum cardinality of a generating set of the vector space. A subset U of a semigroup is said to be independent if every element ofU is not in the subsemigroup generated by the remaining elements of U, i.e., ∀a ∈ U, a /∈ U \ {a} . It can be observed that the minimum size of a generating set need not be equal to the maximum size of an independent set in a semigroup. Accordingly, Howie and Ribeiro have considered various concepts of ranks for a finite semigroup (cf. [5,6]). 1. r1( ) = max{k: every subset U of cardinality k in is independent}. 2. r2( ) = min{|U| : U ⊆ , U = }. 3. r3( ) = max{|U| : U ⊆ , U = ,U is independent}. 4. r4( ) = max{|U| : U ⊆ ,U is independent}.Item On the Structure of the Commuting Graph of Brandt Semigroups(Springer, 2021) Kumar, Jitender