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The paper gives a systematic characterization of the eigenspaces in a max-t algebra, where t is the Łukasiewicz t-norm. A max-Łukasiewicz fuzzy algebra can be used for the description of the states of discrete-event systems. The states can represent a balance between the resources expended during the run of a system (for example fuel or money). The classification of max-Łukasiewicz eigenspaces is described and illustrated by two- and three-dimensional examples: in this case it is possible to accompany the example with graphs. However, the description of the eigenspaces for higher dimensions is also outlined.