Multiple-valued logic functions represented by TSUM, TPRODUCT, NOT and variables

A class of multiple-valued logic functions (TO-functions, for short) expressed by TSUM, TPRODUCT, NOT, and variables is introduced, where TSUM is defined as min (x+y, p-1) and TPRODUCT is redefined as the product that is derived by applying De Morgan's laws to TSUM. It is shown that a set of TO-functions is not a lattice, and that in ternary logic TSUM can be expressed by Lukasiewicz implication, and NOT and its converse holds. It is known that a set of ternary TO-functions is not complete but complete with constants. Moreover, the set is equivalent to ternary functions satisfying normality and includes a set of B-ternary logic functions. For any radix, it is shown that a set of TO-functions is not complete but compete with constants and that the set includes B-multiple-valued logic functions. Moreover, some speculations of the number of TO-functions for less than ten radixes are derived.<
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