Derived graph from finite BL-algebras

This paper investigates the relationship between finite fields and [Formula: see text]-algebras, showing how any finite field can be transformed to a [Formula: see text]-algebra. In doing so, it extends the class of [Formula: see text]-algebras and deepens the connection between these two algebraic structures. We present notion of reproduced bounded lattices and convert any nonempty set to a bounded lattice. Indeed the reproduced bounded lattices, generate a class of [Formula: see text]-algebras. This study introduces the concept of [Formula: see text]-graphs based on the notion of down set in [Formula: see text]-algebras. This research investigates the conditions for the connectivity of [Formula: see text]-graphs, so we peruse some types of [Formula: see text]-algebras such as Gödel [Formula: see text]-algebra, Archimedean [Formula: see text]-algebra and integral [Formula: see text]-algebra. Characterization of the [Formula: see text]-graphs is fundamental in this research, so we compute the number of edges in some [Formula: see text]-graphs via combinators, and prove that [Formula: see text]-graphs under what conditions are isomorphic to underlying lattices of [Formula: see text]-algebras. Finally, we try to extract [Formula: see text]-algebra from graphs, introduce the notion of graph-[Formula: see text]-algebra, and show that any path graph is a graph-[Formula: see text]-algebra.