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  • 1 University of Pune Department of Mathematics Pune 411 007 India Pune 411 007 India
  • 2 S.S.V.P.S's Late Karmveer Dr. P. R. Ghogrey Science College Department of Mathematics Dhule 424 005 India Dhule 424 005 India
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Abstract  

The concept of `adjunct' operation of two lattices with respect to a pair of elements is introduced. A structure theorem namely, `A finite lattice is dismantlable if and only if it is an adjunct of chains' is obtained. Further it is established that for any adjunct representation of a dismantlable lattice the number of chains as well as the number of times a pair of elements occurs remains the same. If a dismantlable lattice L has n elements and n+k edges then it is proved that the number of irreducible elements of L lies between n-2k-2 and n-2. These results are used to enumerate the class of lattices with exactly two reducible elements, the class of lattices with n elements and upto n+1 edges, and their subclasses of distributive lattices and modular lattices.

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