الفهرس 
A Free MonoidsOnly 14 pages are availabe for public view
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Abstract
In [2] the notion of an ”-semigroup (epsilon semigroup) is de ned as a
semigroup S with a unary operation ” : S ! S; x ! ”x that satis es a set
of axioms, extending the notion of an identity in a monoid. The necessarily
unique element ”x (may be also written x”) associated with x 2 S is called
the partial identity of x. A partial monoid is de ned as an ”-semigroup in
which every ”x is central, characterized as a strong semilattice of monoids
and proved to be embedded in a certain partial monoid whose elements are
partial mappings.
In [3] the strong semilattice of the free monoids over the non empty nite
subsets of a set A is de ned and proved to be the free object in the category of
partial monoids and homomorphisms of partial monoids, denoted FPM (A),
and called the free partial monoid on A. This notion is used to develop
a theory of a generalized code, called a partial code, in analogy with the
corresponding classical theory.
With the usual notion of a ”language” as a subset of a free monoid A
on a set A, a subset of free partial monoid on a could be regarded as a
language in the ordinary sense (unless A is a singleton). In that, a subset of
FPM(A) is actually a (disjoint) union of sets, which are subset of (di¤erent)
free monoids (on di¤erent alphabets). Whence a subset of FPM(A) might
be viewed as a generalized language which could not be recognized by an
ordinary automaton.
With the aim of developing a ”generalized” automata (machines) that
could recognize(accepted) generalized language in a free partial monoid on
a set A ,the two structures (machines) so called a ”perfectly generalized au-
tomaton over A” and has been rst created in [1] and characterized as a
”strong semilattice of automata over A”. Languages of these two (equiv-
alent) machines have been de ned in a rather general setting and called
P-languages. Some algebra of P-languages has been developed in analogy
with the corresponding properties in the classical automata-language theory.
The general concept of P recognizability de ned in [1] has laid to establish
the most motivated result, that is, if M is a strong semilattice of automata
over the non empty nite subsets of A, then the union of the language of the
maximal automata in M is a (possibly proper) subset of the P-language
of M. Evidently more deeper result concerning the new developed machine.