X Xx Let be a mapping from set X to then denotes the image of directed graph is specified in the form Journal of Mathematical Sciences Mathematics Education 3 be a dep ID: 878895
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1 X , (X denotes its cardinality x Let
X , (X denotes its cardinality x Let be a mapping from set X to then denotes the image of directed graph is specified in the form Journal of Mathematical Sciences & Mathematics Education 3 be a dependence alphabet and let ~ be a partition of satisfies the conditions: (1) If ns: (1) If v is a clique then ][~66 vv (2) If ) If v is a discrete subgraph then and (3) For all ~ ~ , ~ vuvu ~ } ~ , ~ if and only if there exists an ~ y ~ such t
2 hat Then, ~ is a dependence alphabet.
hat Then, ~ is a dependence alphabet. Therefore, the following concept is well defined: Definition 2.3 A dependence alphabet is called Ehrenfeucht-Rozenberg ER-Reduced, if and only if ~ The Structure of for ER-Reduced Dependence Alphabets The key result of this section provides an algebraic characterization of ER-reduced dependence alphabet by means of its homomorphism monoid, namely, a dependence alphabet is ER-reduced if and only if is unretractive, i.e., To prove this main r
3 esult, we need several lemmas. Lemma 3.1
esult, we need several lemmas. Lemma 3.1 [(Ehrenfaucht & Rozenberg, 1987); Lemma 1.3] Let be a dependence alphabet and let be disjoint clans of Then, for all and all )(},{21GEvu if and only if is called completeis a clique. Lemma 3.2 Let and H be dependence alphabets and let then is a complete clan of for each complete of We first prove that is a clan of Indeed, if then \)()(KHVw We deduce that That is to say, is a clan. Next, we show that is a clique. In fact,
4 for all we have Since K is a clique
for all we have Since K is a clique of we know that is Journal of Mathematical Sciences & Mathematics Education 5 [(Ehrenfaucht & Rozenberg, 1987); Lemma 2.7] Let dependence alphabet. Then ~ is a maximal partition of into complete clans. That is to say, if is an arbitrary partition of into complete clans, then is a refinement of ~ Lemma 3.7 Let be a dependence alphabet, then the mapping ~ defined by )]([~ is a monomorphism from ~ to
5 where, is an arbitrary but fixed set of
where, is an arbitrary but fixed set of representatives of the partition ~ By definition of ~ Evvji ~ ,]{[~~66 if and only if there exist ivu and andjvv such that If j i by assumption, we have ][~~66zjivv By Lemma 3.1, we have if and only if This is equivalent to, ~ ,]{[~~66 if and only if Therefore, is a homomorphism. According to Lemmas 3.2 and 3.6, is one-to-one. Now, we are ready to prove our main result. A dependence alphabet is ER-reduced
6 if and only if is ER-reduced, i.e., the
if and only if is ER-reduced, i.e., then Assume that then there exists a By the finiteness of must not be one-to-one. Hence, there exists such that According to Corollary 3.3, is a complete clan. Thus, is a partition of into complete clans, which contradicts to the maximal property of On the other hand, let be an arbitrary dependence alphabet that satisfies the condition then is ER-reduced. In fact, assume that ~ then based on Lemmas 3.4 and 3.7, there is an epimorp