A New Metatheorem and Subdirect Product Theorem for L-Subgroups

This paper is a continuation of the work of Tom Head ‘Metatheorem for deriving fuzzy theorems from crisp versions‘. The concept of natural extension is introduced which is then applied in the development of a new metatheorem in $L$-setting, where $L$ is a complete chain. The application of this theorem is demonstrated through the notions of generated $L$-subgroups and commutator $L$-subgroups. Moreover, a new subdirect product theorem is developed wherein it is demonstrated that for a group G, the $L$-subgroup lattice can be represented as a subdirect product of copies of its associated lattice of crisp subgroups. The significance of this theorem is exhibited by applying it to a characterization of generalized cyclic groups in terms of the distributivity of the lattice of L-subgroups of a group.