Geodesic
Lattices with Doubly Irreducible Elements
Canadian mathematical bulletin, Tome 17 (1974) no. 1, pp. 91-95

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An element x in a lattice L is join-reducible (meet-reducible) in L if there exist y, z∈L both distinct from x such that x=y⋁z (x=y⋀z); x is join-irreducible (meet-irreducible) in L if it is not join-reducible (meet-reducible) in L; x is doubly irreducible in L if it is both join- and meet-irreducible in L. Let J(L), M(L), and Irr(L) denote the set of all join-irreducible elements in L, meet-irreducible elements in L, and doubly irreducible elements in L, respectively, and l(L) the length of L, that is, the order of a maximum-sized chain in L minus one. 
Lattices with Doubly Irreducible Elements. Canadian mathematical bulletin, Tome 17 (1974) no. 1, pp. 91-95. doi: 10.4153/CMB-1974-016-3
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     title = {Lattices with {Doubly} {Irreducible} {Elements}},
     journal = {Canadian mathematical bulletin},
     pages = {91--95},
     year = {1974},
     volume = {17},
     number = {1},
     doi = {10.4153/CMB-1974-016-3},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1974-016-3/}
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