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 reducts of affine near-semirings over Brandt semigroups(Springer, 2016-09) Kumar, RajeshWe investigate the rank properties of the semigroup reducts of the affine near-semiring A+ (Bn) over the Brandt semigroup Bn.We determine the small, lower, intermediate and large ranks of the additive semigroup reduct An, and find a lower bound for the upper rank of An. In case n ≥ 6, we show that the lower bound is actually equal to the upper rank. We also find the small, lower, and large ranks of the multiplicative semigroup reduct Mn, and provide lower bounds for the intermediate and upper ranks of Mn.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