I think that I would splurge a little on memory and use a hash (or maybe an array if the names are really '1' .. '9'). All the operations you want are easy and straightforward with a hash:

my %state;
@state{1..9}=();
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1. have name, clear bit

   $state{$name} = undef;
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2. have name(s), set bit(s)

   @state{@names} = (1) x @names;
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3. get count of set bits

   my $count = grep {$_} values %state;
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4. have one set bit, get its name

   my $setname = (grep {$state{$_}} keys %state)[0];
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5. have multiple set bits, get names (debugging only)

   my @setnames = grep {$state{$_}} keys %state;
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6. bit-wise logical ops, OR, AND, ...

   # same as ordinary perl
   print $foo if $state{$one} && $state{$tother};
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Fixed typo in #2, thanks blokhead++

For my own tastes, I would use integers and masks, in the "C" style.  Of course, my background is in "C", so you might call me biased.

For example, here's a simple example of clear_bit and set_bit using a mask and a bit index:

#!/usr/bin/perl -w

# Strict
use strict;
use warnings;

# Test code
my $mask = 0b11001011;
clear_bit($mask, 1);
clear_bit($mask, 2);
clear_bit($mask, 3);
clear_bit($mask, 4);

$mask = 0b11001000;
set_bit($mask, 1);
set_bit($mask, 2);
set_bit($mask, 3);
set_bit($mask, 4);

# Subroutines
sub clear_bit {
    my ($mask, $bit_index) = @_;
    printf "TFD> mask 0b%08lb\n", $mask;
    $mask &= ~(1 << ($bit_index - 1));
    printf "TFD> mask 0b%08lb\n\n", $mask;
}

sub set_bit {
    my ($mask, $bit_index) = @_;
    printf "TFD> mask 0b%08lb\n", $mask;
    $mask |= (1 << ($bit_index - 1));
    printf "TFD> mask 0b%08lb\n\n", $mask;
}
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The lines containing "TFD" are temporary debugging statements which can be removed once you've verified that the code is doing what you want.

Of course, if you want to use names in place of the bit_index, you could either pass the string (character) containing the name "1" - "9", and convert it back to an integer, or (where necessary) you could pass a hash containing the names (bit indices) you wish to operate on.

I'm sure you can write the rest of the subroutines, if this is the route you decide to go.  (But feel free to ask if you need more help or suggestions!)