MV-algebras are categorically equivalent to a class of 𝒟ℛ l 1 ( i ) -semigroups

Jiří Rachůnek

Mathematica Bohemica (1998)

  • Volume: 123, Issue: 4, page 437-441
  • ISSN: 0862-7959

Abstract

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In the paper it is proved that the category of -algebras is equivalent to the category of bounded -semigroups satisfying the identity 1 - ( 1 - x ) = x . Consequently, by a result of D. Mundici, both categories are equivalent to the category of bounded commutative -algebras.

How to cite

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Rachůnek, Jiří. "MV-algebras are categorically equivalent to a class of $\mathcal {DR}l_{1(i)}$-semigroups." Mathematica Bohemica 123.4 (1998): 437-441. <http://eudml.org/doc/248317>.

@article{Rachůnek1998,
abstract = {In the paper it is proved that the category of -algebras is equivalent to the category of bounded -semigroups satisfying the identity $1-(1-x)=x$. Consequently, by a result of D. Mundici, both categories are equivalent to the category of bounded commutative -algebras.},
author = {Rachůnek, Jiří},
journal = {Mathematica Bohemica},
keywords = {categorical equivalence; bounded -algebra; -algebra; -semigroup; MV-algebra; -semigroup; categorical equivalence; bounded BCK-algebra},
language = {eng},
number = {4},
pages = {437-441},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {MV-algebras are categorically equivalent to a class of $\mathcal \{DR\}l_\{1(i)\}$-semigroups},
url = {http://eudml.org/doc/248317},
volume = {123},
year = {1998},
}

TY - JOUR
AU - Rachůnek, Jiří
TI - MV-algebras are categorically equivalent to a class of $\mathcal {DR}l_{1(i)}$-semigroups
JO - Mathematica Bohemica
PY - 1998
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 123
IS - 4
SP - 437
EP - 441
AB - In the paper it is proved that the category of -algebras is equivalent to the category of bounded -semigroups satisfying the identity $1-(1-x)=x$. Consequently, by a result of D. Mundici, both categories are equivalent to the category of bounded commutative -algebras.
LA - eng
KW - categorical equivalence; bounded -algebra; -algebra; -semigroup; MV-algebra; -semigroup; categorical equivalence; bounded BCK-algebra
UR - http://eudml.org/doc/248317
ER -

References

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  2. C. C. Chang, A new proof of the completeness of the Lukasiewicz axioms, Trans. Amer. Math. Soc. 93 (1959), 74-80. (1959) Zbl0093.01104MR0122718
  3. R. Cignoli, Free lattice-ordered abelian groups and varieties of MV-algebras, Proc. IX. Latin. Amer. Symp. Math. Logic, Part 1, Not. Log. Mat. 38 (1993), 113-118. (1993) Zbl0827.06012MR1332526
  4. K. Iséki, S. Tanaka, An introduction to the theory of BCK-algebras, Math. Japonica. 23 (1978), 1-26. (1978) MR0500283
  5. T. Kovář, A general theory of dually residuated lattice ordered monoids, Thesis, Palacky Univ. Olomouc, 1996. (1996) 
  6. T. Kovář, Two remarks on dually residuated lattice ordered semigroups, Math. Slovaca. To appear. MR1804468
  7. F. Lacava, Some properties of L-algebras and existencially closed L-algebras, Boll. Un. Mat. Ital., A(5) 16 (1979), 360-366. (In Italian.) (1979) MR0541775
  8. D. Mundici, 10.1016/0022-1236(86)90015-7, J. Funct. Analys. 65 (1986), 15-63. (1986) MR0819173DOI10.1016/0022-1236(86)90015-7
  9. D. Mundici, MV-algebras are categorically equivalent to bounded commutative BCK-algebras, Math. Japonica 31 (1986), 889-894. (1986) Zbl0633.03066MR0870978
  10. J. Rachůnek, 10.1023/A:1022801907138, Czechoslovak Math. J. 123 (1998), 365-372. (1998) MR1624268DOI10.1023/A:1022801907138
  11. K. L. N. Swamy, 10.1007/BF01360284, Math. Ann. 159 (1965), 105-114. (1965) Zbl0138.02104MR0183797DOI10.1007/BF01360284
  12. S. Tanaka, On Λ -commutative algebras, Math. Sem. Notes Kobe 3 (1975), 59-64. (1975) Zbl0324.02053MR0419222
  13. T. Traczyk, On the variety of bounded commutative BCK-algebras, Math. Japonica 24 (1979), 283-292. (1979) Zbl0422.03038MR0550212
  14. H. Yutani, Quasi-commutative BCK-algebras and congruence relations, Math. Sem. Notes Kobe 5 (1977), 469-480. (1977) Zbl0375.02053MR0498112

Citations in EuDML Documents

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  1. Jiří Rachůnek, Filip Švrček, MV-algebras with additive closure operators
  2. Magdalena Harlenderová, Jiří Rachůnek, Modal operators on MV-algebras
  3. Jiří Rachůnek, Ordered prime spectra of bounded D R l -monoids
  4. Jiří Rachůnek, Filip Švrček, Interior and closure operators on bounded commutative residuated l-monoids
  5. Jan Kühr, Representable dually residuated lattice-ordered monoids
  6. Jiří Rachůnek, Dana Šalounová, Classes of filters in generalizations of commutative fuzzy structures
  7. Jan Kühr, Remarks on ideals in lower-bounded dually residuated lattice-ordered monoids
  8. Jan Kühr, Finite-valued dually residuated lattice-ordered monoids
  9. Jiří Rachůnek, Dana Šalounová, Modal operators on bounded commutative residuated ℓ -monoids
  10. Milan Jasem, On lattice-ordered monoids

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