On subalgebra lattices of a finite unary algebra. II.

Konrad Pióro

Mathematica Bohemica (2001)

  • Volume: 126, Issue: 1, page 171-181
  • ISSN: 0862-7959

Abstract

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We use graph-algebraic results proved in [8] and some results of the graph theory to characterize all pairs 〈 𝐋 1 , 𝐋 2 〉 of lattices for which there is a finite partial unary algebra such that its weak and strong subalgebra lattices are isomorphic to 𝐋 1 and 𝐋 2 , respectively. Next, we describe other pairs of subalgebra lattices (weak and relative, etc.) of a finite unary algebra. Finally, necessary and sufficient conditions are found for quadruples 〈 𝐋 1 , 𝐋 2 , 𝐋 3 , 𝐋 4 〉 of lattices for which there is a finite unary algebra having its weak, relative, strong subalgebra and initial segment lattices isomorphic to 𝐋 1 , 𝐋 2 , 𝐋 3 , 𝐋 4 , respectively.

How to cite

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Pióro, Konrad. "On subalgebra lattices of a finite unary algebra. II.." Mathematica Bohemica 126.1 (2001): 171-181. <http://eudml.org/doc/248842>.

@article{Pióro2001,
abstract = {We use graph-algebraic results proved in [8] and some results of the graph theory to characterize all pairs $\langle \mathbf \{L\}_\{1\},\mathbf \{L\}_\{2\}\rangle $ of lattices for which there is a finite partial unary algebra such that its weak and strong subalgebra lattices are isomorphic to $\mathbf \{L\}_\{1\}$ and $\mathbf \{L\}_\{2\}$, respectively. Next, we describe other pairs of subalgebra lattices (weak and relative, etc.) of a finite unary algebra. Finally, necessary and sufficient conditions are found for quadruples $\langle \mathbf \{L\}_\{1\},\mathbf \{L\}_\{2\}, \mathbf \{L\}_\{3\},\mathbf \{L\}_\{4\}\rangle $ of lattices for which there is a finite unary algebra having its weak, relative, strong subalgebra and initial segment lattices isomorphic to $\mathbf \{L\}_\{1\},\mathbf \{L\}_\{2\}, \mathbf \{L\}_\{3\},\mathbf \{L\}_\{4\}$, respectively.},
author = {Pióro, Konrad},
journal = {Mathematica Bohemica},
keywords = {graph; finite unary algebra; partial algebra; subalgebras; subalgebra lattices; graph; finite unary algebra; partial algebra; subalgebras; subalgebra lattices},
language = {eng},
number = {1},
pages = {171-181},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On subalgebra lattices of a finite unary algebra. II.},
url = {http://eudml.org/doc/248842},
volume = {126},
year = {2001},
}

TY - JOUR
AU - Pióro, Konrad
TI - On subalgebra lattices of a finite unary algebra. II.
JO - Mathematica Bohemica
PY - 2001
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 126
IS - 1
SP - 171
EP - 181
AB - We use graph-algebraic results proved in [8] and some results of the graph theory to characterize all pairs $\langle \mathbf {L}_{1},\mathbf {L}_{2}\rangle $ of lattices for which there is a finite partial unary algebra such that its weak and strong subalgebra lattices are isomorphic to $\mathbf {L}_{1}$ and $\mathbf {L}_{2}$, respectively. Next, we describe other pairs of subalgebra lattices (weak and relative, etc.) of a finite unary algebra. Finally, necessary and sufficient conditions are found for quadruples $\langle \mathbf {L}_{1},\mathbf {L}_{2}, \mathbf {L}_{3},\mathbf {L}_{4}\rangle $ of lattices for which there is a finite unary algebra having its weak, relative, strong subalgebra and initial segment lattices isomorphic to $\mathbf {L}_{1},\mathbf {L}_{2}, \mathbf {L}_{3},\mathbf {L}_{4}$, respectively.
LA - eng
KW - graph; finite unary algebra; partial algebra; subalgebras; subalgebra lattices; graph; finite unary algebra; partial algebra; subalgebras; subalgebra lattices
UR - http://eudml.org/doc/248842
ER -

References

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  2. Lectures on Algebras, Equations and Partiality, Rosselló F. (ed.), Technical report B-006, Univ. Illes Balears, Dept. Ciencies Mat. Inf., 1992. (1992) 
  3. Graphs and Hypergraphs, North-Holland, Amsterdam, 1973. (1973) Zbl0254.05101MR0357172
  4. A Model Theoretic Oriented Approach to Partial Algebras, Math. Research Band 32, Akademie Verlag, Berlin, 1986. (1986) Zbl0598.08004MR0854861
  5. Algebraic Theory of Lattices, Prentice Hall Inc., Englewood Cliffs, NJ, 1973. (1973) 
  6. Topics in Universal Algebra, Lecture Notes in Mathemathics 250, Springer, Berlin, 1972. (1972) MR0345895
  7. 10.1023/A:1022418818272, Czechoslovak Math. J. 50 (2000), 295–320. (2000) MR1761388DOI10.1023/A:1022418818272
  8. On subalgebra lattices of a finite unary algebra, part I, Math. Bohem. 126 (2001), 161–170. (2001) MR1826479
  9. On a strong property of the weak subalgebra lattice, Algebra Univers. 40 (1998), 477–495. (1998) MR1681837
  10. 10.2307/2303897, Am. Math. Monthly 46 (1939), 281–283. (1939) Zbl0021.35703MR1524589DOI10.2307/2303897

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