13th International Conference on Membrane Computing - MTA Sztaki

T. Mihálydeák, Z. Csajbók
– an mset M over U can be given in the form
{〈a 1 , M(a 1 )〉, 〈a 2 , M(a 2 )〉, . . . , 〈a n , M(a n )〉};
– in membrane computing if M(a) < ∞ for all a ∈ U, the mset M over U can
be represented { by all permutations of the string w:
w =
a M(a k 1 )
k 1
a M(a k 2 )
k 2
. . . a M(a k l
)
k l
, if M is not an empty mset;
λ, otherwise;
where λ is the empty string.
Remark 2. If all a ∈ U have (countable) infinite occurrences in the mset M over
U i.e. M(a) = ∞ for all a ∈ U, then M is denoted by M ∞ .
Remark 3. In the general theory of msets the set U may be infinite. There is no
need to deal with this case in our investigation.
Set–theoretical operations and relations can be generalized for msets. Let M 1
and M 2 be two msets over U.
1. M 1 = M 2 , if M 1 (a) = M 2 (a) for all a ∈ U;
2. M 1 ⊑ M 2 , if M 1 (a) ≤ M 2 (a) for all a ∈ U;
3. (M 1 ⊓ M 2 )(a) = min {M 1 (a), M 2 (a)} for all a ∈ U;
4. if M ⊆ MS(U), then (⊓M)(a) = min{M(a) | M ∈ M} for all a ∈ U;
5. set–type (⊔) and mset–type (⊕) union can be defined:
(a) (M 1 ⊔ M 2 )(a) = max {M 1 (a), M 2 (a)} for all a ∈ U;
(b) if M ⊆ MS