I need to find all the files with the same "string"

Group by "string" ⇒ use a hash.

use List::Util qw( min );

my %grouped_maxes;
while (<DATA>) {
    chomp;
    my ($hash, $amount, $end) = split /_/;
    push @{ $grouped_maxes{$hash} }, $amount
        if defined($end) && $end eq 'max';
}

for my $hash (keys %grouped_maxes) {
    print("$hash: ", min( @{ $grouped_maxes{$hash} } ), "\n");
}

__DATA__
111aaa222ccc324567fed54333221235_1.04
111aaa222ccc324567fed54333221235_1.05_max
111aaa222ccc324567fed54333221235_0.98_min
111aaa222ccc324567fed54333221235_1.02_max
111aaa222ccc324567fed54333221235_0.21
777aaa222ccc324567fed54333221235_1.04
777aaa222ccc324567fed54333221235_1.07_min
777aaa222ccc324567fed54333221235_1.04_max
888aaa222ccc324567fed54333221235_0.21
[download]

Additionally I also need to know if there is no _max file at all in the selected files

Keep track of all the unique hash strings you see, keep track of the unique hash strings you see with a max, then remove those with a max from the whole set.

Remove duplicates ⇒ use a hash.

use List::Util qw( min );

my %hashes;
my %grouped_maxes;
while (<DATA>) {
    chomp;
    my ($hash, $amount, $end) = split /_/;
    ++$hashes{$hash};
    push @{ $grouped_maxes{$hash} }, $amount
        if defined($end) && $end eq 'max';
}

for my $hash (keys %hashes) {
    if (exists($grouped_maxes{$hash})) {
        my $min_max = min( @{ $grouped_maxes{$hash} } );
        print("$hash: $min_max\n");
    } else {
        print("$hash: No max\n");
    }
}

__DATA__
111aaa222ccc324567fed54333221235_1.04
111aaa222ccc324567fed54333221235_1.05_max
111aaa222ccc324567fed54333221235_0.98_min
111aaa222ccc324567fed54333221235_1.02_max
111aaa222ccc324567fed54333221235_0.21
777aaa222ccc324567fed54333221235_1.04
777aaa222ccc324567fed54333221235_1.07_min
777aaa222ccc324567fed54333221235_1.04_max
888aaa222ccc324567fed54333221235_0.21
[download]

777aaa222ccc324567fed54333221235: 1.04
888aaa222ccc324567fed54333221235: No max
111aaa222ccc324567fed54333221235: 1.02
[download]