Special Dual like Numbers and Lattices

In this book the authors introduce a new type of dual numbers called special dual like numbers. These numbers are constructed using idempotents in the place of nilpotents of order two as new element. That is x = a + bg is a special dual like number where a and b are reals and g is a new element such that g2 =g. The collection of special dual like numbers forms a ring. Further lattices are the rich structures which contributes to special dual like numbers. These special dual like numbers x = a + bg; when a and b are positive reals greater than or equal to one we see powers of x diverge on; and every power of x is also a special dual like number, with very large a and b. On the other hand if a and b are positive reals lying in the open interval (0, 1) then we see the higher powers of x may converge to 0. Another rich source of idempotents is the Neutrosophic number I, as I2 = I. We build several types of finite or infinite rings using these Neutrosophic numbers