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Difference between revisions of "Join-irreducible element"

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====References====
 
====References====
* B. A. Davey, H. A. Priestley, ''Introduction to lattices and order'', 2nd ed. Cambridge University Press  (2002) ISBN 978-0-521-78451-1 {{ZBL|1002.06001}}
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* B. A. Davey, H. A. Priestley, ''Introduction to lattices and order'', 2nd ed. Cambridge University Press  (2002) {{ISBN|978-0-521-78451-1}} {{ZBL|1002.06001}}
* Rota, Gian-Carlo (with P. Doubilet, C. Greene, D. Kahaner, A: Odlyzko and R. Stanley) ''Finite operator calculus'' Academic Press (1975) ISBN 0-12-596650-4 {{ZBL|0328.05007}}
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* Rota, Gian-Carlo (with P. Doubilet, C. Greene, D. Kahaner, A: Odlyzko and R. Stanley) ''Finite operator calculus'' Academic Press (1975) {{ISBN|0-12-596650-4}} {{ZBL|0328.05007}}
  
 
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Latest revision as of 15:00, 19 November 2023

in a lattice $L$

An element $a$ which is not the minimum element $0_L$ (if any) and for which $x < a$ and $y < a$ implies $x \vee y < a$. The latter condition is equivalent to $a = x \vee y$ implies $a = x$ or $a = y$.

Dually, a meet-irreducible element $t$ is not the maximum element $1_L$ if any and $t < x,\,t<y \Rightarrow t < x \wedge y$ or $t = x \wedge y \Rightarrow t = x \,\text{or}\, t = y$. An element which is both join-irreducible and meet-irredicuble is doubly-irreducible.

In a lattice satisfying the descending chain condition, in particular for a finite lattice, the join-irreducible elements are those which cover precisely one element; further, every element is a join of join-irreducible elements.

In the lattice of natural numbers ordered by divisibility, the join-irreducible elements are the prime powers.

References

  • B. A. Davey, H. A. Priestley, Introduction to lattices and order, 2nd ed. Cambridge University Press (2002) ISBN 978-0-521-78451-1 Zbl 1002.06001
  • Rota, Gian-Carlo (with P. Doubilet, C. Greene, D. Kahaner, A: Odlyzko and R. Stanley) Finite operator calculus Academic Press (1975) ISBN 0-12-596650-4 Zbl 0328.05007
How to Cite This Entry:
Join-irreducible element. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Join-irreducible_element&oldid=54545