All Solid Varieties of Semirings
by
Klaus Denecke
University of Potsdam, Potsdam, Germany

A variety V is called solid if every identity in V is satisfied as a hyperidentity. All (infinitely many) solid varieties of semirings were determined in [Pol; 99]. We prove: if a nontrivial variety of semirings is solid then it is contained in the interval between the variety of rectangular semirings and the variety of all normal, idempotent, and distributive semirings. Both varieties are solid. The subvariety lattice of the variety of all normal, idempotent, and distributive semirings was investigated in [Pas; 83]. It is a Boolean lattice with ten atoms and ten dual atoms. Checking this lattice we found exactly one more solid varieties of semirings, the subvariety of the variety of all normal idempotent semirings defined by the additional identity (x+y)(y+x) \approx xy + yx). This gives all solid varieties of semirings.

References

[Pol; 99 ] L. PoláK, All solid varieties of semigroups, Journal of Algebra, 219 (1999), 421-436.

[Pas; 83 ] F. Pastijn, Idempotent distributive semirings II, Semigroup Forum, Vol. 26 (1983) 151-166.

Date received: May 31, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caec-36.