Details
| Original language | English |
|---|---|
| Pages (from-to) | 954-971 |
| Number of pages | 18 |
| Journal | Discrete mathematics |
| Volume | 338 |
| Issue number | 6 |
| Publication status | Published - 6 Jun 2015 |
Abstract
The "bottom" of a partially ordered set (poset) Q is the set Ql of its lower bounds (hence, Ql is empty or a singleton). The poset Q is said to be atomic if each element of Q/Ql dominates an atom, that is, a minimal element of Q/Ql. Thus, all finite posets are atomic. We study general closure systems of down-sets (referred to as ideals) in posets. In particular, we investigate so-called m-ideals for arbitrary cardinals m, providing common generalizations of ideals in lattices and of cuts in posets. Various properties of posets and their atoms are described by means of ideals, polars (annihilators) and residuals, defined parallel to ring theory. We deduce diverse characterizations of atomic posets satisfying certain distributive laws, e.g. by the representation of specific ideals as intersections of prime ideals, or by maximality and minimality properties. We investigate non-dense ideals (down-sets having nontrivial polars) and semiprime ideals (down-sets all of whose residuals are ideals). Our results are constructive in that they do not require any set-theoretical choice principles.
Keywords
- Atom Atomic, Distributive, Ideal, Polar, Residual, Semiprime
ASJC Scopus subject areas
- Mathematics(all)
- Theoretical Computer Science
- Mathematics(all)
- Discrete Mathematics and Combinatorics
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In: Discrete mathematics, Vol. 338, No. 6, 06.06.2015, p. 954-971.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Ideals in atomic posets
AU - Erné, Marcel
AU - Joshi, Vinayak
PY - 2015/6/6
Y1 - 2015/6/6
N2 - The "bottom" of a partially ordered set (poset) Q is the set Ql of its lower bounds (hence, Ql is empty or a singleton). The poset Q is said to be atomic if each element of Q/Ql dominates an atom, that is, a minimal element of Q/Ql. Thus, all finite posets are atomic. We study general closure systems of down-sets (referred to as ideals) in posets. In particular, we investigate so-called m-ideals for arbitrary cardinals m, providing common generalizations of ideals in lattices and of cuts in posets. Various properties of posets and their atoms are described by means of ideals, polars (annihilators) and residuals, defined parallel to ring theory. We deduce diverse characterizations of atomic posets satisfying certain distributive laws, e.g. by the representation of specific ideals as intersections of prime ideals, or by maximality and minimality properties. We investigate non-dense ideals (down-sets having nontrivial polars) and semiprime ideals (down-sets all of whose residuals are ideals). Our results are constructive in that they do not require any set-theoretical choice principles.
AB - The "bottom" of a partially ordered set (poset) Q is the set Ql of its lower bounds (hence, Ql is empty or a singleton). The poset Q is said to be atomic if each element of Q/Ql dominates an atom, that is, a minimal element of Q/Ql. Thus, all finite posets are atomic. We study general closure systems of down-sets (referred to as ideals) in posets. In particular, we investigate so-called m-ideals for arbitrary cardinals m, providing common generalizations of ideals in lattices and of cuts in posets. Various properties of posets and their atoms are described by means of ideals, polars (annihilators) and residuals, defined parallel to ring theory. We deduce diverse characterizations of atomic posets satisfying certain distributive laws, e.g. by the representation of specific ideals as intersections of prime ideals, or by maximality and minimality properties. We investigate non-dense ideals (down-sets having nontrivial polars) and semiprime ideals (down-sets all of whose residuals are ideals). Our results are constructive in that they do not require any set-theoretical choice principles.
KW - Atom Atomic
KW - Distributive
KW - Ideal
KW - Polar
KW - Residual
KW - Semiprime
UR - http://www.scopus.com/inward/record.url?scp=84922566780&partnerID=8YFLogxK
U2 - 10.1016/j.disc.2015.01.001
DO - 10.1016/j.disc.2015.01.001
M3 - Article
AN - SCOPUS:84922566780
VL - 338
SP - 954
EP - 971
JO - Discrete mathematics
JF - Discrete mathematics
SN - 0012-365X
IS - 6
ER -