REGULARITY OF CERTAIN SUBSEMIRINGS OF FULL MATRIX SEMIRINGS

Abstract

For an additively commutative semiring with zero S, we let DVn(S)

denote the set of all A ∈ Mn(S) of the form

⎡

⎢

⎢

⎢

⎢

⎣

x

1

0

• ··

0

x

1

0

x

2

• ··

x

2

0

• ·· ··· ··· ··· ···

0

x

2

• ··

x

2

0

x1 0

• ··

0

x

1

⎤

⎥

⎥

⎥ ⎥ ⎦

where Mn(S) is the full n×n matrix semiring over S. We show conditions

for being regular semirings, left regular semirings, right regular semirings

and intra-regular semirings of DVn(S).