The Rings Which Can Be Recovered by Means of the Difference

Ivan Chajda; Filip Švrček

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica (2013)

  • Volume: 52, Issue: 1, page 49-55
  • ISSN: 0231-9721

Abstract

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It is well known that to every Boolean ring can be assigned a Boolean algebra whose operations are term operations of . Then a symmetric difference of together with the meet operation recover the original ring operations of . The aim of this paper is to show for what a ring a similar construction is possible. Of course, we do not construct a Boolean algebra but only so-called lattice-like structure which was introduced and treated by the authors in a previous paper. In particular, we reached interesting results if the characteristic of the ring is either an odd natural number or a power of 2.

How to cite

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Chajda, Ivan, and Švrček, Filip. "The Rings Which Can Be Recovered by Means of the Difference." Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica 52.1 (2013): 49-55. <http://eudml.org/doc/260732>.

@article{Chajda2013,
abstract = {It is well known that to every Boolean ring $\mathcal \{R\}$ can be assigned a Boolean algebra $\mathcal \{B\}$ whose operations are term operations of $\mathcal \{R\}$. Then a symmetric difference of $\mathcal \{B\}$ together with the meet operation recover the original ring operations of $\mathcal \{R\}$. The aim of this paper is to show for what a ring $\mathcal \{R\}$ a similar construction is possible. Of course, we do not construct a Boolean algebra but only so-called lattice-like structure which was introduced and treated by the authors in a previous paper. In particular, we reached interesting results if the characteristic of the ring $\mathcal \{R\}$ is either an odd natural number or a power of 2.},
author = {Chajda, Ivan, Švrček, Filip},
journal = {Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica},
keywords = {Boolean ring; commutative ring; lattice-like structure; difference; Boolean rings; commutative rings; ring operations; lattice-like structures; symmetric difference},
language = {eng},
number = {1},
pages = {49-55},
publisher = {Palacký University Olomouc},
title = {The Rings Which Can Be Recovered by Means of the Difference},
url = {http://eudml.org/doc/260732},
volume = {52},
year = {2013},
}

TY - JOUR
AU - Chajda, Ivan
AU - Švrček, Filip
TI - The Rings Which Can Be Recovered by Means of the Difference
JO - Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica
PY - 2013
PB - Palacký University Olomouc
VL - 52
IS - 1
SP - 49
EP - 55
AB - It is well known that to every Boolean ring $\mathcal {R}$ can be assigned a Boolean algebra $\mathcal {B}$ whose operations are term operations of $\mathcal {R}$. Then a symmetric difference of $\mathcal {B}$ together with the meet operation recover the original ring operations of $\mathcal {R}$. The aim of this paper is to show for what a ring $\mathcal {R}$ a similar construction is possible. Of course, we do not construct a Boolean algebra but only so-called lattice-like structure which was introduced and treated by the authors in a previous paper. In particular, we reached interesting results if the characteristic of the ring $\mathcal {R}$ is either an odd natural number or a power of 2.
LA - eng
KW - Boolean ring; commutative ring; lattice-like structure; difference; Boolean rings; commutative rings; ring operations; lattice-like structures; symmetric difference
UR - http://eudml.org/doc/260732
ER -

References

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  2. Chajda, I., Pseudosemirings induced by ortholattices, Czech. Math. J. 46 (2008), 405–411. (2008) MR1408295
  3. Chajda, I., Eigenthaler, G., A note on orthopseudorings and Boolean quasirings, Österr. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II 207 (1998), 83–94. (1998) Zbl1040.06003MR1749914
  4. Chajda, I., Länger, H., 10.7151/dmgaa.1008, Discuss. Math., Gen. Algebra Appl. 20 (2010), 87–95. (2010) MR1782088DOI10.7151/dmgaa.1008
  5. Chajda, I., Švrček, F., Lattice-like structures derived from rings, In: Proc. of Salzburg Conference (AAA81), Contributions to General Algebra 20, J. Heyn, Klagenfurt, 2011. (2011) MR2908430
  6. Dorninger, D., Länger, H., Ma̧cyński, M., The logic induced by a system of homomorphisms and its various algebraic characterizations, Demonstratio Math. 30 (1997), 215–232. (1997) MR1446613
  7. Dorninger, D., Länger, H., Ma̧cyński, M., On ring-like structures occuring in axiomatic quantum mechanics, Österr. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II 206 (1997), 279–289. (1997) MR1632939
  8. Dorninger, D., Länger, H., Ma̧cyński, M., 10.1016/S0034-4877(00)86390-9, Rep. Math. Phys. 43 (1999), 499–515. (1999) MR1713402DOI10.1016/S0034-4877(00)86390-9
  9. Dorninger, D., Länger, H., Ma̧cyński, M., 10.1023/A:1003646323230, Intern. J. Theor. Phys. 39 (2000), 1015–1026. (2000) MR1779170DOI10.1023/A:1003646323230
  10. Dorninger, D., Länger, H., Ma̧cyński, M., 10.1016/S0034-4877(01)89034-0, Rep. Math. Phys. 47 (2001), 167–176. (2001) MR1836328DOI10.1016/S0034-4877(01)89034-0
  11. Dorninger, D., Länger, H., Ma̧cyński, M., 10.7151/dmgaa.1041, Discuss. Math., Gen. Algebra Appl. 21 (2001), 239–253. (2001) MR1894319DOI10.7151/dmgaa.1041
  12. Länger, H., Generalizations of the corresspondence between Boolean algebras and Boolean rings to orthomodular lattices, Tatra Mt. Math. Publ. 15 (1998), 97–105. (1998) MR1655082

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