Abstract—We consider a finite, irreducible, aperiodic, time homogenous Markov chain on a fuzzy partition and for the resulting aggregated process we study two aspects emerging from the classical theory on hard partitions. The first aspect is lumpability, a technique for recovering from the large state space of a stochastic system. We provide necessary and sufficient conditions for strong lumpability on the transition probabilities of the original chain for the lumped process to have the Markov property. The second aspect is the asymptotic behavior of the lumped chain. The results are compared with those existing in the classical theory of hard partitions. I. INTRODUCTION Markov chains are frequently used as analytic models in the quantitative evaluations of stochastic systems. Examples of their use may be found in diverse areas such as computer, biological, physical and social sciences as well as in business, economics and engineering. Markov’s work in 1907, celebrating nowadays the centennial anniversary, is an indispensable tool of enormous power. The fundamental property characterizing the model, referred to as the Markov property, is that given the present, past and future transitions of the system are independent of each other. The information that is often sought from this model is either the transient or stationary probability of the system being in a given state. When the number of states is small, it is relatively easy to obtain the transient and/or stationary solutions allowing the prediction of the system behavior. However, as models become more complex the process of obtaining these solutions becomes much more difficult. There is also a wide class of situations, where the modeler does not need information about each state of the system but about classes of states only. This leads to the consideration of a new process, to be called the aggregated or lumped, whose states are the state classes of the original Markov chain. The new stochastic process need not to be Markovian. In order to be able to utilize all the power of the Markov chain theory, it is important to be able to claim that for a given initial distribution the aggregated process has the Markov property. The concept of lumpability on hard partitions has been known for a long time [1], [5] - [8], [11] and [12]. Explicit conditions on the transition probabilities in a discrete time Markov chain were given by Kemeny and Snell in 1960, [8]. They distinguish between strong lumpability when the process is lumpable for any initial distribution on the state Ioannis I. Gerontidis is with the Department of Information Management, Technological Educational Institution of Kavala, Aghios Loukas, 65405 Kavala, Greece (phone: +30 2510462326; email: igeront@teikav.edu.gr). Stavros P. Kontakos is with the Department of Information Management, Technological Educational Institution of Kavala, Aghios Loukas, 65405 Kavala, Greece (phone: +30 2510241410; email: stkontakos@yahoo.gr). space and weak lumpability when the process is lumpable only for some initial probability distributions. Mathematical models of physical processes try to quantify relationships among the variables governing the process. This effort has to cope with three sources of imprecision, that is, inexact measurements, randomness and vagueness resulting to different types of mathematical models known as deterministic, stochastic and fuzzy. The first two models have a long history in scientific era, whereas the third counts forty years of life. It was Zadeh in 1965 [13], who proposed an axiomatic system named “fuzzy set” that attempts to capture and exploit non-statistical uncertainties arising in mathematical models. Since then fuzzy set theory has flourished giving rise to new developments in almost every field of scientific applications. This paper is the first attempt, to the authors’ knowledge, towards a systematic study of Markov chain lumpability on fuzzy partitions. To this end, we analyze a Markov chain model that captures two types of imprecision, i.e. randomness and vagueness. The model was proposed by Bhattacharyya [4], with the aim to study a Markov decision process with fuzzy states. However, the author passed over the important question, when the aggregated process is Markovian. In section II we fix the notation and collect some existing results on partition spaces, the probability of fuzzy events and lumpability on hard partitions. Section III presents the main results, i.e. a characterization of strong lumpability on fuzzy partitions and the asymptotic behavior of the lumped Markov chain, together with a numerical illustration. Finally, in section IV the conclusions are stated. II. PRELIMINARIES This section introduces the basic notation and outlines some existing results necessary for the development of the main results in section III, suitably adapted to our present purposes. By convention, all vectors considered are row vectors. Let [1,...,1] = 1 and [0,...,0] = 0 be the vectors of ones and zeros, i e be a vector with one in the i-th position and zeros elsewhere, and I the identity matrix, the dimension been determined by the context. For a vector [ ] i a = a , let [] diag a be the matrix with elements [ ] i a on the main diagonal and zeros elsewhere. In case that [ ] i a are square matrices instead of scalars, [] diag ⋅ is block diagonal. The transpose of a vector or a matrix is indicated by the transpose operator []' ⋅ . Markov Chain Lumpability on Fuzzy Partitions Ioannis I. Gerontidis and Stavros P. Kontakos 1-4244-1210-2/07/$25.00 ©2007 IEEE. 326