On subalgebra lattices of a finite unary algebra. I.

Konrad Pióro

Mathematica Bohemica (2001)

  • Volume: 126, Issue: 1, page 161-170
  • ISSN: 0862-7959

Abstract

top
One of the main aims of the present and the next part [15] is to show that the theory of graphs (its language and results) can be very useful in algebraic investigations. We characterize, in terms of isomorphisms of some digraphs, all pairs 〈 𝐀 , 𝐋 〉 , where 𝐀 is a finite unary algebra and L a finite lattice such that the subalgebra lattice of 𝐀 is isomorphic to 𝐋 . Moreover, we find necessary and sufficient conditions for two arbitrary finite unary algebras to have isomorphic subalgebra lattices. We solve these two problems in the more general case of partial unary algebras. In the next part [15] we will use these results to describe connections between various kinds of lattices of (partial) subalgebras of a finite unary algebra.

How to cite

top

Pióro, Konrad. "On subalgebra lattices of a finite unary algebra. I.." Mathematica Bohemica 126.1 (2001): 161-170. <http://eudml.org/doc/248834>.

@article{Pióro2001,
abstract = {One of the main aims of the present and the next part [15] is to show that the theory of graphs (its language and results) can be very useful in algebraic investigations. We characterize, in terms of isomorphisms of some digraphs, all pairs $\langle \mathbf \{A\},\mathbf \{L\}\rangle $, where $\mathbf \{A\}$ is a finite unary algebra and $L$ a finite lattice such that the subalgebra lattice of $\mathbf \{A\}$ is isomorphic to $\mathbf \{L\}$. Moreover, we find necessary and sufficient conditions for two arbitrary finite unary algebras to have isomorphic subalgebra lattices. We solve these two problems in the more general case of partial unary algebras. In the next part [15] we will use these results to describe connections between various kinds of lattices of (partial) subalgebras of a finite unary algebra.},
author = {Pióro, Konrad},
journal = {Mathematica Bohemica},
keywords = {unary algebra; partial algebra; subalgebra lattice; directed graph; finite unary algebra; finite unary algebra; partial algebra; subalgebra lattice; directed graph},
language = {eng},
number = {1},
pages = {161-170},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On subalgebra lattices of a finite unary algebra. I.},
url = {http://eudml.org/doc/248834},
volume = {126},
year = {2001},
}

TY - JOUR
AU - Pióro, Konrad
TI - On subalgebra lattices of a finite unary algebra. I.
JO - Mathematica Bohemica
PY - 2001
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 126
IS - 1
SP - 161
EP - 170
AB - One of the main aims of the present and the next part [15] is to show that the theory of graphs (its language and results) can be very useful in algebraic investigations. We characterize, in terms of isomorphisms of some digraphs, all pairs $\langle \mathbf {A},\mathbf {L}\rangle $, where $\mathbf {A}$ is a finite unary algebra and $L$ a finite lattice such that the subalgebra lattice of $\mathbf {A}$ is isomorphic to $\mathbf {L}$. Moreover, we find necessary and sufficient conditions for two arbitrary finite unary algebras to have isomorphic subalgebra lattices. We solve these two problems in the more general case of partial unary algebras. In the next part [15] we will use these results to describe connections between various kinds of lattices of (partial) subalgebras of a finite unary algebra.
LA - eng
KW - unary algebra; partial algebra; subalgebra lattice; directed graph; finite unary algebra; finite unary algebra; partial algebra; subalgebra lattice; directed graph
UR - http://eudml.org/doc/248834
ER -

References

top
  1. Category Theory for Computing Science, Series in Computer Science, Prentice Hall International, London, 1990. (1990) MR1094561
  2. Weak subalgebra lattices, Comment. Math. Univ. Carolin. 31 (1990), 405–410. (1990) Zbl0711.08007MR1078473
  3. Lectures on Algebras, Equations and Partiality, Rosselló F. (ed.), Technical report B-006, Univ. Illes Balears, Dept. Ciencies Mat. Inf., 1992. (1992) 
  4. Graphs and Hypergraphs, North-Holland, Amsterdam, 1973. (1973) Zbl0254.05101MR0357172
  5. A Model Theoretic Oriented Approach to Partial Algebras, Math. Research Band 32, Akademie Verlag, Berlin, 1986. (1986) Zbl0598.08004MR0854861
  6. 10.1017/S0004972700020918, Bull. Austral. Math. Soc. 28 (1983), 247–254. (1983) MR0729011DOI10.1017/S0004972700020918
  7. 10.1016/0022-4049(84)90075-6, J. Pure Appl. Algebra 31 (1984), 47–54. (1984) MR0738204DOI10.1016/0022-4049(84)90075-6
  8. 10.1016/0022-4049(85)90037-4, J. Pure Appl. Algebra 35 (1985), 151–155. (1985) MR0775467DOI10.1016/0022-4049(85)90037-4
  9. A survey of multi-unary algebras, Mimeographed seminar notes, U.C. Berkeley, 1967. (1967) 
  10. Topics in Universal Algebra, Lecture Notes in Mathemathics 250, Springer-Verlag, 1972. (1972) MR0345895
  11. 10.1016/0022-4049(93)90067-4, J. Pure Appl. Algebra 87 (1993), 37–49. (1993) MR1222175DOI10.1016/0022-4049(93)90067-4
  12. 10.1007/BF01197131, Arch. Math. 46 (1986), 18–19. (1986) MR0829806DOI10.1007/BF01197131
  13. 10.1007/BF01182454, Algebra Universalis 27 (1990), 220–229. (1990) MR1037863DOI10.1007/BF01182454
  14. 10.1023/A:1022418818272, Czechoslovak Math. J. 50 (2000), 295–320. (2000) MR1761388DOI10.1023/A:1022418818272
  15. On subalgebra lattices of a finite unary algebra, Part II, Math. Bohem. 126 (2001), 171–181. (2001) MR1826479
  16. 10.4153/CJM-1962-035-1, Canad. J. Math. 14 (1962), 451–460. (1962) MR0137666DOI10.4153/CJM-1962-035-1
  17. 10.1007/BF01188381, Algebra Universalis 24 (1987), 36–40. (1987) MR0921528DOI10.1007/BF01188381
  18. 10.1007/BF01229981, Algebra Universalis 25 (1988), 334–364. (1988) MR0969156DOI10.1007/BF01229981

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.