Geodesic
Ideal extensions of lattices
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 2 (2009), pp. 46-64.

Voir la notice de l'article provenant de la source Math-Net.Ru

Following the well known Schreier's extension of groups, the (ideal) extension of semigroups (without order) have been first considered by A. H. Clifford in Trans. Amer. Math. Soc. 68 (1950), with a detailed exposition of the theory in the monographs of Clifford–Preston and Petrich. The main theorem of the ideal extensions of ordered semigroups has been considered by Kehayopulu and Tsingelis in Comm. Algebra 31 (2003). It is natural to examine the same problem for lattices. Following the ideal extensions of ordered semigroups, in this paper we give the main theorem of the ideal extensions of lattices. Exactly as in the case of semigroups (ordered semigroups), we approach the problem using translations. We start with a lattice $L$ and a lattice $K$ having a least element, and construct (all) the lattices $V$ which have an ideal $L'$ which is isomorphic to $L$ and the Rees quotient $V|L'$ is isomorphic to $K$. Conversely, we prove that each lattice which is an extension of $L$ by $K$ can be so constructed. An illustrative example is given at the end.
Keywords: translation, inner translation, (ideal) extension of a lattice. 
@article{IVM_2009_2_a3,
     author = {N. Kehayopulu},
     title = {Ideal extensions of lattices},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {46--64},
     publisher = {mathdoc},
     number = {2},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2009_2_a3/}
}
TY  - JOUR
AU  - N. Kehayopulu
TI  - Ideal extensions of lattices
JO  - Izvestiâ vysših učebnyh zavedenij. Matematika
PY  - 2009
SP  - 46
EP  - 64
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IVM_2009_2_a3/
LA  - ru
ID  - IVM_2009_2_a3
ER  - 
%0 Journal Article
%A N. Kehayopulu
%T Ideal extensions of lattices
%J Izvestiâ vysših učebnyh zavedenij. Matematika
%D 2009
%P 46-64
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IVM_2009_2_a3/
%G ru
%F IVM_2009_2_a3
N. Kehayopulu. Ideal extensions of lattices. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 2 (2009), pp. 46-64. http://geodesic.mathdoc.fr/item/IVM_2009_2_a3/

[1] Clifford A. H., “Extensions of semigroups”, Trans. Amer. Math. Soc., 68 (1950), 165–173 | DOI | MR

[2] Clifford A. H., Preston G. B., The Algebraic Theory of Semigroups, V. I, Mathematical Surveys, No 7, American Math. Soc., Providence, RI, 1977, 224 pp. | MR

[3] Petrich M., Introduction to semigroups, Merrill Research and Lecture Series, Charles E. Merrill Publishing Co., Columbus, Ohio, 1973, 198 pp. | MR | Zbl

[4] Hulin A. J., “Extensions of ordered semigroups”, Semigroup Forum, 2:4 (1971), 336–342 | DOI | MR | Zbl

[5] Hulin A. J., “Extensions of ordered semigroups”, Czechoslovak Math. J., 26:1 (1976), 1–12 | MR | Zbl

[6] Christoph Fr. T. (Jr.), “Ideal extensions of topological semigroups”, Canad. J. Math., 22:6 (1970), 1168–1175 | MR | Zbl

[7] Hildebrant J. A., “Ideal extensions of compact reductive semigroups”, Semigroup Forum, 25:3–4 (1982), 283–290 | DOI | MR | Zbl

[8] Kehayopulu N., “Ideal extensions of ordered sets”, Int. J. Math. and Math. Sci., 2004, no. 53–56, 2847–2861 | DOI | MR | Zbl

[9] Kehayopulu N., Ponizovskii J. S., Shum K. P., “Retract extensions of ordered sets”, Zap. nauchn. sem. POMI, 321, POMI, SPb., 2005, 205–212 | MR | Zbl

[10] Kehayopulu N., Shum K. P., “Equivalent extensions of ordered sets”, Algebras Groups Geom., 20:4 (2003), 387–402 | MR | Zbl

[11] Kehayopulu N., Tsingelis M., “The ideal extensions of ordered semigroups”, Comm. Algebra, 31:10 (2003), 4939–4969 | DOI | MR | Zbl

[12] Kehayopulu N., Kiriakuli P., “The ideal extensions of lattices”, Simon Stevin, 64:1 (1990), 51–60 | MR | Zbl

[13] Szász G., “Die Translationen der Halbverbände”, Acta Sci. Math., 17 (1956), 165–169 | MR | Zbl

[14] Szász G., Szendrei J., “Über die Translationen der Halbverbände”, Acta Sci. Math., 18 (1957), 44–47 | MR | Zbl

[15] Szász G., “Translationen der Verbände”, Acta Fac. Rer. Nat. Univ. Comenianae Math., 5 (1961), 449–453 | MR | Zbl

[16] Kolibiar M., “Bemerkungen über Translationen der Verbände”, Acta Fac. Rer. Nat. Univ. Comenianae Math., 5 (1961), 455–458 | MR | Zbl

[17] Birkhoff G., Lattice Theory, Third edition, American Math. Soc. Colloq., 25, American Math. Soc. Colloquium Publ., Providence, RI, 1967, 418 pp. | MR | Zbl