M-dwarfs in extremely wide binary systems are very rare , and may thus have different formation processes from those found as single stars or close binaries in the field .
In this paper we search for close companions to a new sample of 36 extremely wide M-dwarf binaries , covering a spectral type range of M1 to M5 and a separation range of 600 - 6500 AU .
We discover 10 new triple systems and one new quadruple system .
We carefully account for selection effects including proper motion , magnitude limits , the detection of close binaries in the SDSS , and other sample biases .
The bias-corrected total high-order-multiple fraction is { 45 } ^ { +18 } _ { -16 } % and the bias-corrected incidence of quadruple systems is < 5 % , both statistically compatible with that found for the more common close M-dwarf multiple systems .
Almost all the detected companions have similar masses to their primaries , although two very low mass companions , including a candidate brown dwarf , are found at relatively large separations .
We find that the close-binary separation distribution is strongly peaked towards < 30AU separations .
There is marginally significant evidence for a change in high-order M-dwarf multiplicity with binding energy and total mass .
We also find 2 \sigma evidence of an unexpected increased high-order-multiple fraction for the widest targets in our survey , with a high-order-multiple fraction of { 21 } ^ { +17 } _ { -7 } % for systems with separations up to 2000AU , compared to { 77 } ^ { +9 } _ { -22 } % for systems with separations > 4000AU .
These results suggest that the very widest M-dwarf binary systems need higher masses to form or to survive .