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In mathematics, an empty product, or nullary product or vacuous product, is the result of multiplying no factors. It is by convention equal to the multiplicative identity (assuming there is an identity for the multiplication operation in question), just as the empty sum—the result of adding no numbers—is by convention zero, or the additive identity.
Since the empty category is vacuously a discrete category, a terminal object can be thought of as an empty product (a product is indeed the limit of the discrete diagram {X i}, in general). Dually, an initial object is a colimit of the empty diagram 0 → C and can be thought of as an empty coproduct or categorical sum.
This representation is commonly extended to all positive integers, including 1, by the convention that the empty product is equal to 1 (the empty product corresponds to k = 0). This representation is called the canonical representation [10] of n, or the standard form [11] [12] of n. For example, 999 = 3 3 ×37, 1000 = 2 3 ×5 3, 1001 = 7×11×13.
Second, and empty product is not the result of anything, it is something like "0!" whose value is 1, but it is not identical to 1 (or otherwise conversely "1 is an empty product", which seems a bad formulation). In short, one should make distinction between expressions and there values, and an empty product is an expression.
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