In this paper, we generalize Clifford semirings to left Clifford semirings by means of the so-called band semirings. We also discuss a special case of this kind of semirings, that is, strong distributive lattices of left rings.

Y Q GUO

K P SHUM

M K SEN

University of Calcutta (informally known as Calcutta University or CU) is a collegiate public state university located in Kolkata, West Bengal, India.&nbsp;Within India it is recognized as a &quot;Five-Star University&quot; and accredited &quot;A&quot; Grade by National Assessment and Accreditation Council.

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The University of Calcutta has secured the Fifth position among the Universities of India in the prestigious &quot;Indian University Ranking 2019&quot; list, released by the NIRF of the Ministry of Human Resource Development, Government of India.

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It was established on 24 January 1857, and was one of the first institutions in Asia to be established as a multidisciplinary and Western-style university. Its alumni and faculty include several heads of state, heads of government, social reformers, prominent artists, only academy award winner and Dirac medal winner in India, many Fellows of the Royal Society and five Nobel laureates as of 2019 which is highest in South Asia.

University Of Calcutta

Acta Mathematica Sinica, English Series

In this paper we introduce the notion of almost idempotent semirings as the semirings with semilattice additive reduct satisfying the identity x + x 2 = x2, and characterize eight subclasses of the variety AI of all almost idempotent semirings corresponding to the eight subvarieties of the variety Nℬ 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, P f(F) is freely generated by the band F in the variety AI. This semiring Pf(F) is free in a subclass of A I if and only if F is in the corresponding subvariety of Nℬ. © 2010 Academy of Mathematics and Systems Science, Chinese Academy of Sciences, and Suzhou University.

Algebra Colloquium

The structure of almost idempotent semirings

In [1] Bandelt and Petrich described the class of inverse semirings which arise as subdirect products of a distributive lattice and a ring. In this paper, we characterize all semirings which are subdirect products of a distributive lattice and a ring. For this we introduce the concepts of a strong distributive lattice of semirings, an E-inversive semiring and a Clifford semiring and show that a semiring S is a subdirect product of a distributive lattice and a ring if and only if S is an E-inversive strong distributive lattice of halfrings. Further a Clifford semiring which is, in fact, an inversive subdirect product of a distributive lattice and a ring, is characterized as a strong distributive lattice of rings. Finally, as a consequence of these results we extend a result of Galbiati and Veronesi [2] in the case of Boolean semirings.

Journal	Data powered by TypesetActa Mathematica Sinica, English Series
Publisher	Data powered by TypesetSpringer Verlag
ISSN	1439-8516