L-algebras with additional conditions

This paper delves into the properties of L-algebras, particularly those satisfying additional conditions like (K), (C), (D), and $\left(D_s\right)$. We see that new condition $\left(D_s\right)$ is equivalent to (C). Then we introduce and investigate special L-algebras and provides a full representation of this structure. Furthermore, we define and investigate linear ideals in L-algebras, showing their relationship to prime ideals in L-algebras satisfying (K) and (D). We also explore the representability of L-algebras, presenting a necessary and sufficient condition for representability in terms of a specific inequality, when the L-algebra satisfies $\left(D_s\right)$. Finally, we investigate ordinal sums of L-algebras, which allows for the construction of new L-algebras, including linearly ordered ones, and analyze its properties in relation to representability.