Classification of L-algebras up to order 4

L-algebras, connected to algebraic logic and quantum structures, were first introduced by Rump [13]. In [5], the number of some finite L-algebras is counted. But classifying and finding examples of L-algebras is not always an easy task and is very important. So, in this paper, we classify the set of all non-isomorphic L-algebras up to order 4. For this, we define the notion of partial and total conditions and by using them, we characterize all of the L-algebras of orders 2, 3 and 4. We prove that that there are 5 L-algebras of order 3 and 44 L-algebras of order 4, up to isomorphism.