Extension hoops and states

We first introduce a new kind of algebras named left Ehoop, in which the top element “1” is changed as one of the maximal elements. Therefore, left Ehoops are generalizations of left hoops, which is one of the most fundamental fuzzy logic algebras as the semantical system of fuzzy logic system. Additionally, we explore the connections between left Ehoops and several related structures, including EL-algebras, CL-algebras, BCK-algebras, and quantum B-algebras. To further clarify the link between left Ehoops and EL-algebras, we also define the concept of self-similar EL-algebra. More importantly, we give an equivalent characterization such that left Ehoops become self-similar EL-algebras. Lastly, we define Bosbach states, Riečan states, and state-morphisms on left Ehoops, while examining their key properties and interrelationships. Our findings show that the collection of Bosbach states on a left Ehoop constitutes a compact Hausdorff space, and the mapping $\phi$ from state-measures to Bosbach states is an affine homeomorphism.