.png){kind=icon}
.png){kind=icon}
Return
Positional Numeral Systems over Polyadic Rings
.png){kind=icon}
PRE
AI
DOI：10.3390/math14091530.png)
.png){kind=icon}
Original request for help.png){kind=icon}
Share.png){kind=icon}
Save
Abstract
En 中文
We construct positional numeral systems that work natively over nonderived polyadic m , n -rings whose addition takes m arguments and multiplication takes n. In such rings, the length of an admissible additive word and a multiplicative tower are not arbitrary (as in the binary case) but "quantized". Our main contributions are the following. Existence: Every commutative m , n -ring admits a base-p place-value expansion that respects the word length constraint in terms of numbers of operation compositions & ell;(mult) = & ell;(add) ( m - 1 ) + 1 . Lower bound: The minimum number of digits is greater than or equal to the arity of addition m. Representability gap: For m , n >= 3 only a proper subset of ring elements possesses finite expansions, characterized by congruence-class arity shape invariants I-(m) and J((n)) . Mixed-base "polyadic clocks": Allowing a different base at each position enlarges the design space quadratically in the digit count. Catalogs: Explicit tables for the integer rings Z(4,3) and Z(6,5) illustrate how ordinary integers lift to distinct polyadic variables. These results lay the groundwork for faster arity-aware arithmetic, exotic coding schemes, and hardware that exploits operations beyond the binary pair.
Keywords:
positional number system
arity
polyadic structure
polyadic number
congruence class
numeral
polyadic ring
mixed base
querelement