# Multiplicatively idempotent semirings

Ivan Chajda; Helmut Länger; Filip Švrček

Mathematica Bohemica (2015)

- Volume: 140, Issue: 1, page 35-42
- ISSN: 0862-7959

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topChajda, Ivan, Länger, Helmut, and Švrček, Filip. "Multiplicatively idempotent semirings." Mathematica Bohemica 140.1 (2015): 35-42. <http://eudml.org/doc/269881>.

@article{Chajda2015,

abstract = {Semirings are modifications of unitary rings where the additive reduct does not form a group in general, but only a monoid. We characterize multiplicatively idempotent semirings and Boolean rings as semirings satisfying particular identities. Further, we work with varieties of enriched semirings. We show that the variety of enriched multiplicatively idempotent semirings differs from the join of the variety of enriched unitary Boolean rings and the variety of enriched bounded distributive lattices. We get a characterization of this join.},

author = {Chajda, Ivan, Länger, Helmut, Švrček, Filip},

journal = {Mathematica Bohemica},

keywords = {semiring; commutative semiring; multiplicatively idempotent semiring; semiring of characteristic 2; simple semiring; unitary Boolean ring; bounded distributive lattice},

language = {eng},

number = {1},

pages = {35-42},

publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},

title = {Multiplicatively idempotent semirings},

url = {http://eudml.org/doc/269881},

volume = {140},

year = {2015},

}

TY - JOUR

AU - Chajda, Ivan

AU - Länger, Helmut

AU - Švrček, Filip

TI - Multiplicatively idempotent semirings

JO - Mathematica Bohemica

PY - 2015

PB - Institute of Mathematics, Academy of Sciences of the Czech Republic

VL - 140

IS - 1

SP - 35

EP - 42

AB - Semirings are modifications of unitary rings where the additive reduct does not form a group in general, but only a monoid. We characterize multiplicatively idempotent semirings and Boolean rings as semirings satisfying particular identities. Further, we work with varieties of enriched semirings. We show that the variety of enriched multiplicatively idempotent semirings differs from the join of the variety of enriched unitary Boolean rings and the variety of enriched bounded distributive lattices. We get a characterization of this join.

LA - eng

KW - semiring; commutative semiring; multiplicatively idempotent semiring; semiring of characteristic 2; simple semiring; unitary Boolean ring; bounded distributive lattice

UR - http://eudml.org/doc/269881

ER -

## References

top- Chajda, I., Švrček, F., Lattice-like structures derived from rings, Contributions to General Algebra 20, Proceedings of the 81st Workshop on General Algebra Salzburg, Austria Johannes Heyn Klagenfurt (2012), 11-18 J. Czermak et al. (2012) Zbl1321.06011MR2908430
- Chajda, I., Švrček, F., 10.7151/dmgaa.1181, Discuss. Math., Gen. Algebra Appl. 31 (2011), 175-184. (2011) Zbl1262.06005MR2953910DOI10.7151/dmgaa.1181
- Clouse, D. J., Guzmán, F., 10.1007/s00012-011-0102-y, Algebra Univers. 64 (2010), 231-249. (2010) Zbl1217.06007MR2781078DOI10.1007/s00012-011-0102-y
- Golan, J. S., Semirings and Affine Equations over Them: Theory and Applications, Mathematics and Its Applications 556 Kluwer Academic Publishers, Dordrecht (2003). (2003) Zbl1042.16038MR1997126
- Golan, J. S., Semirings and Their Applications, Kluwer Academic Publishers Dordrecht (1999). (1999) Zbl0947.16034MR1746739
- Grätzer, G., Lakser, H., Płonka, J., 10.4153/CMB-1969-095-x, Can. Math. Bull. 12 (1969), 741-744. (1969) Zbl0188.04903MR0276160DOI10.4153/CMB-1969-095-x
- Guzmán, F., 10.1016/0022-4049(92)90108-R, J. Pure Appl. Algebra 78 (1992), 253-270. (1992) Zbl0770.16020MR1163278DOI10.1016/0022-4049(92)90108-R
- Jedlička, P., The rings which are Boolean, Part II, Acta Univ. Carol., Math. Phys. 53 (2012), 73-75. (2012) MR3099402

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