Strong quasi k-ideals and the lattice decompositions of semirings with semilattice additive reduct

2014 ◽  
Vol 34 (1) ◽  
pp. 27 ◽  
Author(s):  
Anjan Kumar Bhuniya ◽  
Kanchan Jana
Keyword(s):  
Additive Reduct

scholarly journals Semigroup of k-bi-Ideals of a Semiring with Semilattice Additive Reduct

2016 ◽  
Vol 49 (2) ◽  
Author(s):  
A. K. Bhuniya ◽  
K. Jana
Keyword(s):  
Regular Semigroup ◽  
Normal Band ◽  
Additive Reduct

AbstractWe associate a semigroup B(S) to every semiring S with semilattice additive reduct, namely the semigroup of all k-bi-ideals of S; and such semirings S have been characterized by this associated semigroup B(S). A semiring S is k-regular if and only if B(S) is a regular semigroup. For the left k-Clifford semirings S, B(S) is a left normal band; and consequently, B(S) is a semilattice if S is a k-Clifford semiring. Also we show that the set B

Idempotent semirings with a commutative additive reduct

2001 ◽  
Vol 64 (2) ◽  
pp. 289-296 ◽  
Author(s):  
Xianzhong Zhao
Keyword(s):  
Idempotent Semirings ◽  
Additive Reduct

Expanding the additive reduct of a model of Peano arithmetic

2003 ◽  
Vol 49 (4) ◽  
pp. 363-368
Author(s):  
Masahiko Murakami ◽  
Akito Tsuboi
Keyword(s):  
Peano Arithmetic ◽  
Additive Reduct

scholarly journals THE ZELEZNIKOW PROBLEM ON A CLASS OF ADDITIVELY IDEMPOTENT SEMIRINGS

2013 ◽  
Vol 95 (3) ◽  
pp. 404-420
Author(s):  
YONG SHAO ◽  
SINIŠA CRVENKOVIĆ ◽  
MELANIJA MITROVIĆ
Keyword(s):  
Semigroup Forum ◽  
Inverse Semigroup ◽  
Regular Ring ◽  
Clifford Semigroup ◽  
Binary Operations ◽  
Idempotent Semirings ◽  
Additive Reduct

AbstractA semiring is a set $S$ with two binary operations $+ $ and $\cdot $ such that both the additive reduct ${S}_{+ } $ and the multiplicative reduct ${S}_{\bullet } $ are semigroups which satisfy the distributive laws. If $R$ is a ring, then, following Chaptal [‘Anneaux dont le demi-groupe multiplicatif est inverse’, C. R. Acad. Sci. Paris Ser. A–B 262 (1966), 274–277], ${R}_{\bullet } $ is a union of groups if and only if ${R}_{\bullet } $ is an inverse semigroup if and only if ${R}_{\bullet } $ is a Clifford semigroup. In Zeleznikow [‘Regular semirings’, Semigroup Forum 23 (1981), 119–136], it is proved that if $R$ is a regular ring then ${R}_{\bullet } $ is orthodox if and only if ${R}_{\bullet } $ is a union of groups if and only if ${R}_{\bullet } $ is an inverse semigroup if and only if ${R}_{\bullet } $ is a Clifford semigroup. The latter result, also known as Zeleznikow’s theorem, does not hold in general even for semirings $S$ with ${S}_{+ } $ a semilattice Zeleznikow [‘Regular semirings’, Semigroup Forum 23 (1981), 119–136]. The Zeleznikow problem on a certain class of semirings involves finding condition(s) such that Zeleznikow’s theorem holds on that class. The main objective of this paper is to solve the Zeleznikow problem for those semirings $S$ for which ${S}_{+ } $ is a semilattice.

The Structure of Almost Idempotent Semirings

2010 ◽  
Vol 17 (spec01) ◽  
pp. 851-864 ◽  
Author(s):  
M. K. Sen ◽  
A. K. Bhuniya
Keyword(s):  
Distributive Lattice ◽  
Idempotent Semiring ◽  
Idempotent Semirings ◽  
Additive Reduct

In this paper we introduce the notion of almost idempotent semirings as the semirings with semilattice additive reduct satisfying the identity x + x2 = x2, and characterize eight subclasses of the variety [Formula: see text] of all almost idempotent semirings corresponding to the eight subvarieties of the variety [Formula: see text] of all normal bands. Every almost idempotent semiring S is a distributive lattice of rectangular almost idempotent semirings. Given a semigroup F, the semiring Pf(F) of all finite non-empty subsets of F is almost idempotent precisely when F is a band, and in this case, Pf(F) is freely generated by the band F in the variety [Formula: see text]. This semiring Pf(F) is free in a subclass of [Formula: see text] if and only if F is in the corresponding subvariety of [Formula: see text].

On semirings whose additive reduct is a semilattice

2010 ◽  
Vol 82 (1) ◽  
pp. 131-140 ◽  
Author(s):  
M. K. Sen ◽  
A. K. Bhuniya
Keyword(s):  
Additive Reduct

scholarly journals Additively orthodox semirings with special transversals

2021 ◽  
Vol 7 (3) ◽  
pp. 4153-4167
Author(s):  
Kaiqing Huang ◽  
◽  
Yizhi Chen ◽  
Miaomiao Ren ◽  
◽  
...  
Keyword(s):  
Orthodox Semigroup ◽  
Completely Regular ◽  
Additive Reduct

<abstract><p>A semiring $ (S, +, \cdot) $ is called additively orthodox semiring if its additive reduct $ (S, +) $ is a orthodox semigroup. In this paper, by introducing some special semiring transversals as the tools, the constructions of additively orthodox semirings with a skew-ring transversal or with a generalized Clifford semiring transversal are established. Meanwhile, it is shown that an additively orthodox semiring with a generalized Clifford semiring transversal is a b-lattice of additively orthodox semirings with skew-ring transversals. Consequently, the corresponding results of Clifford semirings and generalized Clifford semirings in reference (M. K. Sen, S. K. Maity, K. P. Shum, Clifford semirings and generalized Clifford semirings, Taiwan. J. Math., 9 (2005), 433–444) and completely regular semirings in reference (S. K. Maity, M. K. Sen, K. P. Shum, On completely regular semirings, Bull. Cal. Math. Soc., 98 (2006), 319–328) are extended and strengthened.</p></abstract>

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