semimodular lattice
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Order ◽  
2020 ◽  
Author(s):  
Gábor Czédli ◽  
Robert C. Powers ◽  
Jeremy M. White

AbstractLet L be a lattice of finite length and let d denote the minimum path length metric on the covering graph of L. For any $\xi =(x_{1},\dots ,x_{k})\in L^{k}$ ξ = ( x 1 , … , x k ) ∈ L k , an element y belonging to L is called a median of ξ if the sum d(y,x1) + ⋯ + d(y,xk) is minimal. The lattice L satisfies the c1-median property if, for any $\xi =(x_{1},\dots ,x_{k})\in L^{k}$ ξ = ( x 1 , … , x k ) ∈ L k and for any median y of ξ, $y\leq x_{1}\vee \dots \vee x_{k}$ y ≤ x 1 ∨ ⋯ ∨ x k . Our main theorem asserts that if L is an upper semimodular lattice of finite length and the breadth of L is less than or equal to 2, then L satisfies the c1-median property. Also, we give a construction that yields semimodular lattices, and we use a particular case of this construction to prove that our theorem is sharp in the sense that 2 cannot be replaced by 3.



2020 ◽  
Vol 39 (3) ◽  
pp. 2831-2843
Author(s):  
Peng He ◽  
Xue-Ping Wang

Let D be a finite distributive lattice with n join-irreducible elements. It is well-known that D can be represented as the congruence lattice of a rectangular lattice L which is a special planer semimodular lattice. In this paper, we shall give a better upper bound for the size of L by a function of n, improving a 2009 result of G. Grätzer and E. Knapp.



Studia Logica ◽  
2017 ◽  
Vol 106 (4) ◽  
pp. 691-698
Author(s):  
Peng He ◽  
Xue-ping Wang
Keyword(s):  


Author(s):  
K.Nith ya ◽  
◽  
V.Vino ba
Keyword(s):  


Order ◽  
2015 ◽  
Vol 33 (2) ◽  
pp. 231-237 ◽  
Author(s):  
Gábor Czédli


Order ◽  
2011 ◽  
Vol 29 (1) ◽  
pp. 85-103 ◽  
Author(s):  
Gábor Czédli


2010 ◽  
Vol 324 (9) ◽  
pp. 2089-2111 ◽  
Author(s):  
Peter R. Jones


Order ◽  
1991 ◽  
Vol 8 (2) ◽  
pp. 143-158 ◽  
Author(s):  
Herbert Abels


Author(s):  
D. C. Trueman

SynopsisLet W be a semigroup with W\W2 non-empty, such that if ρ is a congruence on W with xpy for all x, y= W\W2, then zpw for all z, w= W2. We prove that the lattice of congruences on W is directly indecomposable, and conclude that a direct product of cyclic semigroups, with at least two non-group direct factors, has a directly indecomposable lattice of congruences. We find that the lattice of congruences on a direct product S1×S2×V of two non-trivial cyclic semigroups S1 and S2, one not being a group, and any other semigroup V, is not lower semimodular, and hence, not modular. We then prove that any finite ideal extension of a group by a nil semigroup has an upper semimodular lattice of congruences, and conclude that a finite direct product of finite cyclic semigroups has an upper semimodular lattice of congruences.



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