A new lattice construction: The amalgamation of lattices

Source abstract

Abstract Let L and K be two distributive lattices, J be an ideal of K , $K,$ upper K comma and f : L → K $f:L\to K$ f colon upper L right arrow upper K be a lattice homomorphism. In this article, we introduce the sublattice L ⋈ f J := { ( a , f ( a ) ∨ i ) ∣ a ∈ L , i ∈ J } $L \bowtie ^f J:= \{(a,f(a)\vee i) \mid a\in L,\ i\in J \}$ upper L bowtie Superscript f Baseline upper J colon equals left brace left parenthesis a comma f left parenthesis a right parenthesis or i right parenthesis vertical bar a element of upper L comma i element of upper J right brace of L × K $L \times K$ upper L times upper K , called the “amalgamated lattice of L with K along J with respect to f ,” and study some of its basic properties.