Adjoint Semilattice and Minimal Brouwerian Extensions of a Hilbert Algebra

Jānis Cīrulis

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica (2012)

  • Volume: 51, Issue: 2, page 41-51
  • ISSN: 0231-9721

Abstract

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Let A : = ( A , → , 1 ) be a Hilbert algebra. The monoid of all unary operations on A generated by operations α p : x ↦ ( p → x ) , which is actually an upper semilattice w.r.t. the pointwise ordering, is called the adjoint semilattice of A . This semilattice is isomorphic to the semilattice of finitely generated filters of A , it is subtractive (i.e., dually implicative), and its ideal lattice is isomorphic to the filter lattice of A . Moreover, the order dual of the adjoint semilattice is a minimal Brouwerian extension of A , and the embedding of A into this extension preserves all existing joins and certain “compatible” meets.

How to cite

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Cīrulis, Jānis. "Adjoint Semilattice and Minimal Brouwerian Extensions of a Hilbert Algebra." Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica 51.2 (2012): 41-51. <http://eudml.org/doc/246296>.

@article{Cīrulis2012,
abstract = {Let $A := (A,\rightarrow ,1)$ be a Hilbert algebra. The monoid of all unary operations on $A$ generated by operations $\alpha _p\colon x \mapsto (p \rightarrow x)$, which is actually an upper semilattice w.r.t. the pointwise ordering, is called the adjoint semilattice of $A$. This semilattice is isomorphic to the semilattice of finitely generated filters of $A$, it is subtractive (i.e., dually implicative), and its ideal lattice is isomorphic to the filter lattice of $A$. Moreover, the order dual of the adjoint semilattice is a minimal Brouwerian extension of $A$, and the embedding of $A$ into this extension preserves all existing joins and certain “compatible” meets.},
author = {Cīrulis, Jānis},
journal = {Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica},
keywords = {adjoint semilattice; Brouwerian extension; closure endomorphism; compatible meet; filter; Hilbert algebra; implicative semilattice; subtraction; adjoint semilattice; Brouwerian extension; closure endomorphism; compatible meet; Hilbert algebra; implicative semilattice; subtraction; finitely generated filters; lattice of ideals; lattice of filters},
language = {eng},
number = {2},
pages = {41-51},
publisher = {Palacký University Olomouc},
title = {Adjoint Semilattice and Minimal Brouwerian Extensions of a Hilbert Algebra},
url = {http://eudml.org/doc/246296},
volume = {51},
year = {2012},
}

TY - JOUR
AU - Cīrulis, Jānis
TI - Adjoint Semilattice and Minimal Brouwerian Extensions of a Hilbert Algebra
JO - Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica
PY - 2012
PB - Palacký University Olomouc
VL - 51
IS - 2
SP - 41
EP - 51
AB - Let $A := (A,\rightarrow ,1)$ be a Hilbert algebra. The monoid of all unary operations on $A$ generated by operations $\alpha _p\colon x \mapsto (p \rightarrow x)$, which is actually an upper semilattice w.r.t. the pointwise ordering, is called the adjoint semilattice of $A$. This semilattice is isomorphic to the semilattice of finitely generated filters of $A$, it is subtractive (i.e., dually implicative), and its ideal lattice is isomorphic to the filter lattice of $A$. Moreover, the order dual of the adjoint semilattice is a minimal Brouwerian extension of $A$, and the embedding of $A$ into this extension preserves all existing joins and certain “compatible” meets.
LA - eng
KW - adjoint semilattice; Brouwerian extension; closure endomorphism; compatible meet; filter; Hilbert algebra; implicative semilattice; subtraction; adjoint semilattice; Brouwerian extension; closure endomorphism; compatible meet; Hilbert algebra; implicative semilattice; subtraction; finitely generated filters; lattice of ideals; lattice of filters
UR - http://eudml.org/doc/246296
ER -

References

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  1. Cīrulis, J., Multipliers in implicative algebras, Bull. Sect. Log. (Łódź) 15 (1986), 152–158. (1986) Zbl0634.03067MR0907610
  2. Cīrulis, J., Multipliers, closure endomorphisms and quasi-decompositions of a Hilbert algebra, In: Chajda et al., I. (eds) Contrib. Gen. Algebra Verlag Johannes Heyn, Klagenfurt, 2005, 25–34. (2005) Zbl1082.03056MR2166943
  3. Cīrulis, J., 10.2478/s11533-007-0008-2, Centr. Eur. J. Math. 5 (2007), 264–279. (2007) Zbl1125.03047MR2300273DOI10.2478/s11533-007-0008-2
  4. Curry, H. B., Foundations of Mathematical logic, McGraw-Hill, New York, 1963. (1963) Zbl0163.24209MR0148529
  5. Diego, A., Sur les algèbres de Hilbert, Gauthier-Villars; Nauwelaerts, Paris; Louvain, 1966. (1966) Zbl0144.00105MR0199086
  6. Henkin, L., An algebraic characterization of quantifiers, Fund. Math. 37 (1950), 63–74. (1950) Zbl0041.34804MR0040234
  7. Horn, A., 10.2307/2964545, Journ. Symb. Logic 27 (1962), 391–399. (1962) MR0171706DOI10.2307/2964545
  8. Huang, W., Liu, F., 10.1007/BF03325431, Semigroup Forum 58 (1999), 317–322. (1999) Zbl0928.06012MR1678492DOI10.1007/BF03325431
  9. Huang, W., Wang, D., Adjoint semigroups of BCI-algebras, Southeast Asian Bull. Math. 19 (1995), 95–98. (1995) Zbl0859.06016MR1366413
  10. Iseki, K., Tanaka, S., An introduction in the theory of BCK-algebras, Math. Japon. 23 (1978), 1–26. (1978) MR0500283
  11. Karp, C. R., Set representation theorems in implicative models, Amer. Math. Monthly 61 (1954), 523–523 (abstract). (1954) 
  12. Karp, C. R., Languages with expressions of infinite length, Univ. South. California, 1964 (Ph.D. thesis). (1964) Zbl0127.00901MR0176910
  13. Kondo, M., 10.1155/S0161171201010985, Int. J. Math. 28 (2001), 535–543. (2001) Zbl1007.06014MR1895299DOI10.1155/S0161171201010985
  14. Marsden, E. L., 10.1007/BF00650494, J. Philos. Log. 1 (1972), 195–200. (1972) MR0476504DOI10.1007/BF00650494
  15. Schmidt, J., Quasi-decompositions, exact sequences, and triple sums of semigroups I. General theory. II Applications, In:Contrib. Universal Algebra Colloq. Math. Soc. Janos Bolyai (Szeged) 17 North-Holland, Amsterdam, 1977, 365–428. (1977) MR0472657
  16. Tsinakis, C., Brouwerian semilattices determined by their endomorphism semigroups, . Houston J. Math. 5 (1979), 427–436. (1979) Zbl0431.06003MR0559982
  17. Tsirulis, Ya. P., Notes on closure endomorphisms of implicative semilattices, Latvijskij Mat. Ezhegodnik 30 (1986), 136–149 (in Russian). (1986) MR0878277

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