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Results on the use of category theory for the study of lattice-valued finite state machines
Information Sciences 288 (2014) 279–289
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Information Sciences journal homepage: www.elsevier.com/locate/ins
Results on the use of category theory for the study of lattice-valued finite state machines q Jian-Gang Tang a,⇑, Mao-Kang Luo b, Juan Tang c a
College of Mathematics and Statistics, YiLi Normal University, Yining 835000, PR China College of Mathematics, Sichuan University, Chengdu 610064, PR China c College of Electronics and Information Engineering, Sichuan University, Chengdu 610064, PR China b
a r t i c l e
i n f o
Article history: Received 25 June 2006 Received in revised form 27 May 2014 Accepted 23 June 2014 Available online 31 July 2014 Keywords: Category of L-fuzzy finite state machines Category of L-fuzzy quasi-transformation semigroups Category of L-fuzzy quasi-transformation monoids Category of L-fuzzy transformation semigroups Category of L-fuzzy transformation monoids Functor
a b s t r a c t In this paper, we introduce the concepts of category of lattice-valued finite state machines, category of quasi-lattice-valued transformation semigroups, category of quasi-lattice-valued transformation monoids, category of lattice-valued transformation semigroups, category of lattice-valued transformation monoids and six pairs of functors, each pair of functors have the adjoint property, we discuss the relation between the categories with functors and give the adjoint theorems and relation theorems. 2014 Elsevier Inc. All rights reserved.
1. The concept of lattice-valued finite machines In 1967, Wee [1] firstly introduced the fuzzy finite automata, which generated many interesting explorations. Its important applications in learning systems, automatic control, pattern recognition and database, and so on were also studied by many researchers; the more details and the fuzzy finite automata were studied by Wee and Fu [2], Santos [3–6], Lee and Zadeh [7], Kumbhojkar and Chaudhari [8,9], Malik et al. [10–15], and so on; the authors can also refer to [16,17]. In this paper, we introduce the concept of lattice-valued finite machine extended from the concept of fuzzy finite machine in Wee [1]. For this purpose, we firstly introduce the concept of the Heyting algebra and the intuitionistic first-order logic as follow. Definition 1.1. A lattice L is called a Heyting algebra, if for any a; b 2 L, there exists an element ða ! bÞ 2 L such that
8c 2 L;
c 6 ða ! bÞ () c ^ a 6 b:
For any element a in a Heyting algebra L, especially denote :a ¼ ða ! 0Þ, call :a the negation of a. q
Supported by NSFC (Nos. 11161050, 31240020).
⇑ Corresponding author.
E-mail addresses: [email protected] (J.-G. Tang), [email protected] (M.-K. Luo). http://dx.doi.org/10.1016/j.ins.2014.06.027 0020-0255/ 2014 Elsevier Inc. All rights reserved.
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Any lattice ðL; 6Þ gives rise to a category, with the elements of L as objects, and with precisely one morphism from p to q if and only if p 6 q; in other words, the morphisms are pairs ðp; qÞ such that p 6 q, and the composition operation for L is uniquely determined by the transitivity of the order relation 6, and the mapping with order preserving between two complete lattices is the functor between two categories. Let L be a lattice. For every a 2 L; a ^ ðÞ : L ! L is an order-preserving mapping, by Definition 1.1, L is a Heyting algebra if and only if 8a 2 L, the functor
a ^ ðÞ : L ! L has a right adjoint
a ! ðÞ : L ! L: The following conclusion is often used in this paper, and its proof is easy: Theorem 1.1. Let L be a lattice, ? be a binary operation on L. Then ðL; !Þ is a Heyting algebra if and only if the following conditions hold: (1) (2) (3) (4)
a 6 b () a ! b ¼ 1, a ^ ða ! bÞ ¼ a ^ b, b ^ ða ! bÞ ¼ b, a ! ðb ^ aÞ ¼ ða ! bÞ ^ ða ! cÞ.
If L is a complete Heyting algebra, then 8a 2 L; a ! ðÞ : L ! L is an order-preserving mapping, and then, for any b 2 L, we have
a ! b ¼ _fx 2 Lja ^ x 6 bg: Syntax of the intuitionistic first-order logic is similar to the classical first-order logic. We have three primitive connective symbols : (negation), ^ (conjunction), and ? (implication) and a primitive quantifier " (universal quantifier). The connectives _ (disjunction), M (bi-implication) and the existential quantifier $ are defined in terms of :, ^, ?, and " in the usual way. In addition, let L be a Heyting algebra, 2 (membership) be a binary (primitive) predicate symbol. Then # (inclusion) and (equality) can be defined by 2 as usual. The semantics of the intuitionistic first-order logic is given by interpreting the connectives :, ^, and ? as the operations :ðÞ, ^, and ?, respectively, on L and by interpreting the quantifier " as the least upper bound in L. In addition, the truth value of the set-theoretical formula x 2 A is kx 2 Ak ¼ AðxÞ. It is worth indicating that in this paper the set A and its membership function are identified. In the intuitionistic first-order logic, 1 is the unique designated truth value. In other words, a formula u is valid if and only if its truth value kuk is 1. Definition 1.2. A lattice-valued finite state machine (or LFM for short) is defined as a triple M ¼ ðQ ; R; lÞ, where Q is a finite set of internal states; R is a finite set of symbols called the input alphabet; l is a lattice-valued subset of Q R Q , i.e., l is a function from Q R Q to L, where L is a Heyting algebra. If lðp; r; qÞ ¼ a with a 2 L, then p is said to be a previous state, r is said to be input, q is said to be a next state of p and a is said to be a membership degree or probable degree. The purpose of using lattice-valued sets in the finite automata is to interpret an imprecise character of changes in the internal state of the finite automata from one state to another state under an environmental change. It is more general to interpret this imprecision by using lattice-valued sets with a method of inferential flow with the multi-layer and the multi-branch which well be presented here. If lðp; r; qÞ ¼ a, where a 2 L, it can be interpreted that the lattice-valued finite state machine M is in a state p 2 Q and suppose it inputs r 2 R, it will cause the state to change on the membership degree a 2 L to a new state q 2 Q . And we say that the edge
begins at p, ends at q, and carries the label
ðr=aÞðsÞ ¼
a
if
r=a, where r=a can be regarded as a lattice-valued subset of R defined by
s ¼ r;
0 otherwise:
A (crisp) finite state machine ðQ ; R; dÞ, where d : Q R ! Q is a relation, can be considered as a lattice-valued finite state machine ðQ ; R; lÞ, if we define l : Q R Q ! L as follows:
lðp; r; qÞ ¼
1 if dðp; rÞ ¼ q; 0
otherwise:
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On the other hand, a lattice-valued finite state machine ðQ ; R; lÞ, when state machine ðQ ; R; dÞ, if we define
dðp; rÞ ¼ q if and only if
lðQ R Q Þ # f0; 1g, can be regarded as a finite
lðp; r; qÞ ¼ 1
However, in this situation d is not necessarily a partial function, but may happen to be a proper relation. In other words, ðQ ; R; dÞ may not be a deterministic finite state machine. A fuzzy finite state machine M ¼ ðQ ; R; lÞ [1] can also be regarded as a lattice-valued finite state machine, if we set L ¼ ½0; 1. 2. The category of lattice-valued finite state machines Definition 2.1. Let M1 ¼ ðQ 1 ; R1 ; l1 Þ and M2 ¼ ðQ 2 ; R2 ; l2 Þ be two lattice-valued finite state machines. Let a : Q 1 ! Q 2 and b : R1 ! R2 be mappings. A pair (a, b) is called a strong homomorphism, written ða; bÞ : M1 ! M2 , if
l2 ðaðqÞ; bðxÞ; aðpÞÞ ¼ _fl1 ðq; x; tÞjt 2 Q 1 ; aðtÞ ¼ aðpÞg; 8q; p 2 Q 1 and 8x 2 R1 . A strong homomorphism ða; bÞ : M1 ! M2 is called a strong monomorphism (strong epimorphism, or strong isomorphism), if both the mappings a and b are injective (surjective, or bijective).
Definition 2.2. The category of lattice-valued finite state machines (or Lfsm for short) is defined as follows: object: lattice-valued finite state machine M ¼ ðQ ; R; lÞ; morphism: strong epimorphisms ða; bÞ : M1 ! M2 . Let M ¼ ðQ ; R; lÞ be a lattice-valued finite state machine where Q and R are non-empty finite sets and l is a lattice-valued subset of Q R Q ; R is called the alphabet, an element of R is called symbol, Rþ is a set of words over R, and R is Rþ adding the empty word e, i.e. R ¼ Rþ [ feg. It is obvious that R is a free monoid on R and Rþ is a free semigroup on R, respectively, under usual juxtaposition of words. A lattice-valued set l : Q R Q ! L can induce two functions lþ : Q Rþ Q ! L and l : Q R Q ! L defined by
lþ ðp; r; qÞ ¼ lðp; r; qÞ; lþ ðp; xr; qÞ ¼ _flþ ðp; x; rÞ ^ lðr; r; qÞjr 2 Q g; for all p; q 2 Q ; x 2 Rþ ; r 2 R. And l satisfies l jRþ ¼ lþ with
l ðp; e; qÞ ¼
1 if p ¼ q 0
if p – q:
Theorem 2.1. Let M ¼ ðQ ; R; lÞ be a lattice-valued finite state machine. Then
lþ ðp; xy; qÞ ¼
_ flþ ðp; x; rÞ ^ lþ ðr; y; qÞÞjr 2 Qg;
for all p; q 2 Q and x; y 2 Rþ . And
l is the same.
Proof. Let x ¼ a1 ; . . . ; an and y ¼ b1 ; . . . ; bm with a1 ; . . . ; an ; b1 ; . . . ; bm 2 R. Then
_
flþ ðp; x; rÞ ^ lþ ðr; y; qÞÞjr 2 Q g ¼
_
flþ ðp; a1 ; . . . ; an ; rÞ ^ lþ ðr; b1 ; . . . ; bm ; qÞjr 2 Q g _ n_ ¼ flðp; a1 ; q1 Þ ^ ^ lðqn1 ; an ; rÞjq1 ; . . . ; qn1 2 Qg ^
_
o flðr; b1 ; qn Þ ^ ^ lðqnþm1 ; bm ; qÞjqn ; . . . ; qnþm1 2 Q g r 2 Q
_
flðp; a1 ; q1 Þ ^ ^ lðqn1 ; an ; rÞ ^ lðr; b1 ; qn Þ ^ ^lðqnþm ; bm ; qÞÞjq1 ; . . . ; qnþm1 ; r 2 Q
¼
¼ lþ ðp; a1 ; . . . ; an b1 ; . . . ; bm ; qÞ ¼ lþ ðp; xy; qÞ:
þ Let b be a function from a set R1 to a set R2 . Then, bþ is the homomorphism of semigroups from Rþ 1 to R2 defined by
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bþ ðx1 ; . . . ; xn Þ ¼ bðx1 Þ; . . . ; bðxn Þ; for any xi 2 R1 ; i ¼ 1; 2; . . . ; n. And b is the homomorphism of monoids from R1 to R2 defined by
b ðx1 ; . . . ; xn Þ ¼ bðx1 Þ; . . . ; bðxn Þ; for any xi 2 R1 ; i ¼ 1; 2; . . . ; n. Then bþ (b ) is called an extended homomorphism of semigroups (monoids). Theorem 2.2. Let M1 ¼ ðQ 1 ; R1 ; l1 Þ and M2 ¼ ðQ 2 ; R2 ; l2 Þ be lattice-valued finite state machines, ða; bÞ : M1 ! M2 a strong homomorphism. Then
lþ2 ðaðqÞ; bþ ðxÞ; aðpÞÞ ¼ _flþ1 ðq; x; tÞjt 2 Q 1 ; aðtÞ ¼ aðpÞg; for any q; p 2 Q 1 and 8x 2 Rþ 1 . And
l2 ðaðqÞ; b ðxÞ; aðpÞÞ ¼ _fl1 ðq; x; tÞjt 2 Q 1 ; aðtÞ ¼ aðpÞg; for any q; p 2 Q 1 and 8x 2 R1 . Proof. The proof is straightforward. h Definition 2.3. A quasi-lattice-valued transformation semigroup is a triple ðQ ; S; qÞ consisting of a finite set Q, a finite semigroup S and a lattice-valued subset of Q S Q q, i.e., q is a function from Q S Q to L, and W qðp; uv ; qÞ ¼ fqðp; u; rÞ ^ qðr; v ; qÞÞjr 2 Q g for all p; q 2 Q and u; v 2 S. A quasi-lattice-valued transformation semigroup S ¼ ðQ ; S; qÞ is regarded as a lattice-valued finite state machine ðQ ; R; lq Þ by setting R to S and lq ¼ q, we denote it as FSMðSÞ. Definition 2.4. A quasi-lattice-valued transformation semigroup S ¼ ðQ ; S; qÞ is called a quasi-lattice-valued transformation monoid if S is a monoid with an identity e such that the following condition holds:
qðp; e; qÞ ¼
1 if p ¼ q; 0 if p – q:
A quasi-lattice-valued transformation monoid S ¼ ðQ ; S; qÞ can also be regarded as a lattice-valued finite state machine ðQ ; R; lq Þ by putting R to be S and lq ¼ q, we also denote it as FSMðSÞ. Definition 2.5. Let S 1 ¼ ðQ 1 ; S1 ; q1 Þ and S 2 ¼ ðQ 2 ; S2 ; q2 Þ be two quasi-lattice-valued transformation semigroups. Let f : Q 1 ! Q 2 be a mapping and g : S1 ! S2 a homomorphism of semigroups. A pair ðf ; gÞ is called a strong homomorphism, written ðf ; gÞ : S 1 ! S 2 , if
q2 ðf ðqÞ; gðxÞ; f ðpÞÞ ¼ _fq1 ðq; x; tÞjt 2 Q 1 ; f ðtÞ ¼ f ðpÞg; 8q; p 2 Q 1 and 8x 2 S1 . A strong homomorphism ðf ; gÞ : S 1 ! S 2 is called a strong monomorphism (strong epimorphism, or strong isomorphism), if both the mappings a and b are injectives (surjectives, or bijectives). Definition 2.6. The category of quasi-lattice-valued transformation semigroups (or qLts for short) is defined as follows: objects: quasi-lattice-valued transformation semigroup S ¼ ðQ ; S; qÞ; morphism: strong epimorphisms ðf ; gÞ : S 1 ! S 2 . We can also define the category qLtm of quasi-lattice-valued transformation monoids with quasi-lattice-valued transformation monoid as object and the strong epimorphism of quasi-lattice-valued transformation monoids as morphism. A lattice-valued finite state machine M ¼ ðQ ; R; lÞ naturally introduces a quasi-lattice-valued transformation semigroup ðQ ; Rþ ; ql Þ, where ql is defined by
ql ðp; x; qÞ ¼ lþ ðp; x; qÞ: We call ðQ ; Rþ ; ql Þ a quasi-lattice-valued transformation semigroup introduced by M and denote it as qFTSðMÞ, i.e. qFTSðMÞ ¼ ðQ ; Rþ ; ql Þ. Theorem 2.3. Let M ¼ ðQ ; R; lÞ be a lattice-valued finite state machine, then ðQ ; Rþ ; ql Þ given by M is a quasi-lattice-valued transformation semigroup. It is clear that ðQ ; Rþ ; ql Þ is a quasi-lattice-valued transformation semigroup, and then, we can additionally define
ql ðp; e; qÞ ¼
1 if p ¼ q; 0
if p – q:
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Then ðQ ; R ; ql Þ given by M is a quasi-lattice-valued transformation monoid and denote it as qFTMðMÞ, i.e. qFTMðMÞ ¼ ðQ ; R ; ql Þ . Theorem 2.4. Let M ¼ ðQ ; R; lÞ be a lattice-valued finite state machine, then ðQ ; R ; ql Þ introduced by M is a quasi-latticevalued transformation monoid. Theorem 2.5. Let M1 ¼ ðQ 1 ; R1 ; l1 Þ and M2 ¼ ðQ 2 ; R2 ; l2 Þ be two lattice-valued finite state machines and ða; bÞ : M2 ! M2 a strong epimorphism of lattice-valued finite state machines. Then there exists a strong epimorphism of quasi-lattice-valued transformation semigroups (monoids)
ðf a ; g b Þ : qFTSðM1 Þ ! qFTSðM2 Þ; ððf a ; g b Þ : qFTMðM1 Þ ! qFTMðM2 ÞÞ: Proof. The proof is straightforward. h Theorem 2.6. Let S 1 ¼ ðQ 1 ; S1 ; q1 Þ and S 2 ¼ ðQ 2 ; S2 ; q2 Þ be two quasi-lattice-valued transformation semigroups (quasi-latticevalued transformation monoids) and ðf ; gÞ : S 1 ! S 2 a strong epimorphism of quasi-lattice-valued transformation semigroups (quasi-lattice-valued transformation monoids). Then there exists a strong epimorphism of lattice-valued finite state machines
ðaf ; bg Þ : FSMðS 1 Þ ! FSMðS 2 Þ: Proof. Let FSMðS 1 Þ ¼ ðQ 1 ; R1 ; lq1 Þ; FSMðS 2 Þ ¼ ðQ 2 ; R2 ; lq2 Þ and lq1 ¼ q1 ; lq2 ¼ q2 , respectively. Define af : Q 1 ! Q 2 by af ¼ f and bg : R1 ! R2 by bg ¼ g. Then
lq2 ðaf ðqÞ; bg ðxÞ; af ðpÞÞ ¼ q2 ðf ðqÞ; gðxÞ; f ðpÞÞ ¼ _fq1 ðq; x; tÞjt 2 Q 1 ; f ðtÞ ¼ f ðpÞg ¼ _flq1 ðq; x; tÞjt 2 Q 1 ; af ðtÞ ¼ af ðpÞg 8q; p 2 Q 1 and 8x 2 R1 . Hence there exists a strong epimorphism of lattice-valued finite state machines ðaf ; bg Þ : FSMðS 1 Þ ! FSMðS 2 Þ: For a quasi-lattice-valued finite semigroup S = ðQ ; S; qÞ, we can define a relation on S by x y if and only if
qðp; x; qÞ ¼ qðp; y; qÞ for all p; q 2 Q . h Theorem 2.7. Let S ¼ ðQ ; S; qÞ be a quasi-lattice-valued finite semigroup. Then is a congruence relation on S. Proof. Clearly, is an equivalence relation on S, let z2S and x y, then for all W W p; q 2 Q ; qðp; xz; qÞ ¼ fqðp; x; rÞ ^ qðr; z; qÞjr 2 Q g ¼ fqðp; y; rÞ ^ qðr; z; qÞjr 2 Q g ¼ qðp; yz; qÞ by Theorem 3.1, hence xz yz. Similarly, zx zy. Thus is a congruence relation on S. Given a quasi-lattice-valued finite semigroup S ¼ ðQ ; S; qÞ, let ½x ¼ fy 2 Rþ jx yg, where x 2 S and let S= ¼ f½xjx 2 Sg be denoted by Q ðSÞ. h Theorem 2.8. Let S ¼ ðQ ; S; qÞ be a lattice-valued finite semigroup. Then Q ðSÞ is a semigroup, the binary operation on Q ðSÞ is defined by ½x½y ¼ ½xy. Given a quasi-lattice-valued finite monoid S ¼ ðQ ; M; qÞ. We also can define a congruence relation on M, and then, get a monoid M= ¼ f½xjx 2 Mg which is denoted by Q ðSÞ, the binary operation on Q ðSÞ is defined by ½x½y ¼ ½xy. Definition 2.7. A lattice-valued transformation semigroup is a triple ðQ ; S; qÞ consisting of a finite set Q, a finite semigroup S and a lattice-valued subset of Q S Q q, i.e., q is a function from Q S Q to L, and the following conditions hold: W (1) qðp; uv ; qÞ ¼ fqðp; u; rÞ ^ qðr; v ; qÞÞjr 2 Q g for all p; q 2 Q and u; v 2 S; (2) For u; v 2 S, if qðp; u; qÞ ¼ qðp; v ; qÞ for all p; q 2 Q , then u ¼ v . A lattice-valued transformation semigroup S ¼ ðQ ; S; qÞ is regarded as a quasi-lattice-valued finite semigroup immediately. A lattice-valued transformation semigroup S ¼ ðQ ; S; qÞ is regarded as a lattice-valued finite state machine ðQ ; R; lq Þ by setting R to be S and lq ¼ q, we denote it as FSMðSÞ. Definition 2.8. A lattice-valued transformation semigroup S ¼ ðQ ; S; qÞ is called a lattice-valued transformation monoid if S is a monoid with an identity e such that the following condition holds:
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qðp; e; qÞ ¼
1 if p ¼ q; 0 if p – q:
A lattice-valued transformation monoid S ¼ ðQ ; S; qÞ can be regarded as a quasi-lattice-valued finite monoid immediately. A lattice-valued transformation monoid S ¼ ðQ ; S; qÞ can also be regarded as a lattice-valued finite state machine ðQ ; S; lq Þ by setting R to be M and lq ¼ q, we also denote it as FSMðSÞ. Definition 2.9. Let S 1 ¼ ðQ 1 ; S1 ; q1 Þ and S 2 ¼ ðQ 2 ; S2 ; q2 Þ be two lattice-valued transformation semigroups. Let f : Q 1 ! Q 2 be a mapping and g : S1 ! S2 a homomorphism of semigroups. A pair ðf ; gÞ is called a strong homomorphism, written as ðf ; gÞ : S 1 ! S 2 , if
q2 ðf ðqÞ; gðxÞ; f ðpÞÞ ¼ _fq1 ðq; x; tÞjt 2 Q 1 ; f ðtÞ ¼ f ðpÞg; 8q; p 2 Q 1 and 8x 2 S1 . A strong homomorphism ðf ; gÞ : S 1 ! S 2 is called a strong monomorphism (strong epimorphism, or strong isomorphism), if both the mappings a and b are injective (surjective, or bijective). Definition 2.10. The category of lattice-valued transformation semigroups (or Lts for short) is defined as follows objects: lattice-valued transformation semigroup S ¼ ðQ ; S; qÞ; morphism: strong epimorphisms ðf ; gÞ : S 1 ! S 2 . We can also define the strong epimorphism of lattice-valued transformation monoids and the category Ltm of lattice-valued transformation monoids. Now we consider a lattice-valued transformation semigroup and a lattice-valued transformation monoid introduced by a quasi-lattice-valued transformation semigroup and a lattice-valued transformation monoid, respectively. A quasi-lattice-valued transformation semigroup S ¼ ðQ ; S; qÞ naturally introduces a lattice-valued transformation semigroup ðQ ; S= ; q= Þ, where q= is defined by
q= ðp; ½x; qÞ ¼ qðp; x; qÞ for all ½x 2 S= . We call ðQ ; S= ; q= Þ as a lattice-valued transformation semigroup introduced by S and denote it as FTSðSÞ, i.e. FTSðSÞ ¼ ðQ ; S= ; q= Þ. A quasi-lattice-valued transformation monoid S ¼ ðQ ; M; qÞ can also naturally introduces a lattice-valued transformation monoid ðQ ; M= ; q= Þ, it denoted by FTMðSÞ, i.e. FTMðSÞ ¼ ðQ ; S= ; q= Þ. Theorem 2.9. Let S 1 ¼ ðQ 1 ; S1 ; q1 Þ and S 2 ¼ ðQ 2 ; S2 ; q2 Þ be two quasi-lattice-valued transformation semigroups and ðf ; gÞ : S 2 ! S 2 a strong epimorphism of quasi-lattice-valued transformation semigroups. Then there exists a strong epimorphism of lattice-valued transformation semigroups
ðf ; g= Þ : FTSðS 1 Þ ! FTSðS 2 Þ: Proof. Let S1 = ¼ Q ðS 1 Þ; S2 = ¼ Q ðS 2 Þ and suppose s 2 S1 = , then there exists a 2 S1 such that s ¼ ½a, the -equivalence class containing a. Suppose a ¼ r1 ; . . . ; rn ; ri 2 S1 , we define g= ðsÞ ¼ ½gðaÞ0 where ½gðaÞ0 is the 0 -equivalence class containing gðaÞ ¼ gðr1 Þ; . . . ; gðrn Þ 2 S2 . (Note that is induced by S 1 and 0 is induced by S 2 .) We must firstly establish that g= : S1 = ! S2 =0 is well-defined. Suppose that s ¼ ½b, where b ¼ s1 ; . . . ; sm with sj 2 S1 . Now for any q; p 2 Q 1 ; q1 ðq; a; pÞ ¼ q1 ðq; b; pÞ, let q0 ; p0 2 Q 2 , there exists q; p 2 Q 1 such that q0 ¼ f ðqÞ and p0 ¼ f ðpÞ, respectively. Then
q2 ðq0 ; gðaÞ; p0 Þ ¼ q2 ðf ðqÞ; gðaÞ; f ðpÞÞ ¼ _fq1 ðq; a; tÞjt 2 Q 1 ; f ðtÞ ¼ f ðpÞg ¼ _fq1 ðq; b; tÞjt 2 Q 1 ; f ðtÞ ¼ f ðpÞg ¼ q2 ðf ðqÞ; gðbÞ; f ðpÞÞ ¼ q2 ðq0 ; gðbÞ; p0 Þ: Thus, gðaÞ0 gðbÞ and g= is well-defined. Now, let q; p 2 Q ; s ¼ ½a 2 S1 = , then
q2 = ðf ðqÞ; g= ðsÞ; f ðpÞÞ ¼ q2 = ðf ðqÞ; g= ð½aÞ; f ðpÞÞ ¼ q2 = ðf ðqÞ; ½gðaÞ0 ; f ðpÞÞ ¼ q2 ðf ðqÞ; gðaÞ; f ðpÞÞ ¼ _fq1 ðq; a; tÞjt 2 Q 1 ; f ðtÞ ¼ f ðpÞg ¼ _fq1 = ðq; ½a; tÞjt 2 Q 1 ; f ðtÞ ¼ f ðpÞg ¼ _fq1 = ðq; s; tÞjt 2 Q 1 ; f ðtÞ ¼ f ðpÞg
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where s ¼ ½a and a 2 S1 . So ðf ; g= Þ : FTSðS 1 Þ ! FTSðS 2 Þ is a strong epimorphism of lattice-valued transformation semigroups. Theorem 2.10. Let S 1 ¼ ðQ 1 ; S1 ; q1 Þ and S 2 ¼ ðQ 2 ; S2 ; q2 Þ be two quasi-lattice-valued transformation monoids and ðf ; gÞ : S 2 ! S 2 a strong epimorphism of quasi-lattice-valued transformation monoids. Then there exists a strong epimorphism of lattice-valued transformation monoids
ðf ; g= Þ : FTMðS 1 Þ ! FTMðS 2 Þ: Proof. The proof is similar to Theorem 3.9.
h
Theorem 2.11. Let S 1 ¼ ðQ 1 ; S1 ; q1 Þ and S 2 ¼ ðQ 2 ; S2 ; q2 Þ be two lattice-valued transformation semigroups and ðf ; gÞ : S 1 ! S 2 a strong epimorphism of lattice-valued transformation semigroups. Then there exists a strong epimorphism of quasi-lattice-valued transformation semigroups (quasi-lattice-valued transformation monoids).
ðf ; gÞ : S 1 ! S 2 : Proof. The proof is straightforward. h
3. Main results Category theory is a very general tool that can define, analyze, compare, etc. at a macro (very general) level different concepts and properties. Automata, including the lattice-valued finite state machines, are very general tools that can be formalized in many different ways, applied in many different fields, serve for so different purposes, etc. Hence, the use of categorytheoretical tools for the analysis and synthesis of automata, including the lattice-valued finite state machines, makes much sense. Our main results obtained in this paper can be briefly summarized as: From a macro (very general) point of view, a latticevalued finite state machines is equivalent (in terms of its behavior) to a lattice-valued transformation semigroup, and we can use well-known results from the theory of semigroups to study the lattice-valued finite state machines. This is a very important result because the theory of semigroups is a general and powerful apparatus, and it may give us many tools. Consider the diagram of the categories and the functors.
Diagram 3.1 In Diagram 3.1, Lfsm is the category of lattice-valued finite state machines, qLts is the category of quasi-lattice-valued transformation semigroups, qLtm is the category of quasi-lattice-valued transformation monoids, Lts is the category of lattice-valued transformation semigroups, and Ltm is the category of lattice-valued transformation monoids. Extþ and Ext are the extending functors, Tr þ and Tr are the transforming functors, Emp and Ems are the embedding functors. These functors are defined as follows: The extending functor Extþ : Lsm ! qLts from the category Lfsm of lattice-valued finite state machines to the category qLts of quasi-lattice-valued transformation semigroups is defined as follows: For all M ¼ ðQ ; R; lÞ 2 ObðLfsmÞ; Extþ ðMÞ ¼ qFTSðMÞ ¼ ðQ ; Rþ ; lþ Þ 2 ObðqLtsÞ is a quasi-lattice-valued transformation semigroup introduced by M. For all strong epimorphism of lattice-valued finite state machines ða; bÞ : ðQ 1 ; R1 ; l1 Þ ! ðQ 2 ; R2 ; l2 Þ; Extþ ðða; bÞÞ ¼ þ þ þ ðf a ; g b Þ : ðQ 1 ; Rþ 1 ; l1 Þ ! ðQ 2 ; R2 ; l2 Þ is a strong epimorphism of quasi-lattice-valued transformation semigroups according to Theorem 2.5. The extending functor Ext : Lfsm ! qLtm from the category Lfsm of lattice-valued finite state machines to the category qLtm of quasi-lattice-valued transformation monoids is defined as follows:
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For all M ¼ ðQ ; R; lÞ 2 ObðLfsmÞ, Ext ðMÞ ¼ qFTMðMÞ ¼ ðQ ; R ; l Þ 2 ObðqLtmÞ is a quasi-lattice-valued transformation monoid introduced by M. For all strong epimorphism of lattice-valued finite state machines ða; bÞ : ðQ 1 ; R1 ; l1 Þ ! ðQ 2 ; R2 ; l2 Þ; Ext ðða; bÞÞ ¼ ðf a ; g b Þ : ðQ 1 ; R1 ; l1 Þ ! ðQ 2 ; R2 ; l2 Þ is a strong epimorphism of quasi-lattice-valued transformation monoids according to Theorem 2.5. Other functors can be similarly defined. Consider the diagram of categories and the functors.
Diagram 3.2 U s and U p are the forgetful functors, S and Sþ are the shifting functors, R and Rþ are the restricting functors. These functors are defined as follows: The forgetful functor U s : Ltm ! Lts from the category Ltm of lattice-valued transformation monoids to the category Lts of lattice-valued transformation semigroups maps a lattice-valued transformation monoid ðQ ; M; qÞ to the underlying lattice-valued transformation semigroup ðQ ; S; qÞ where S ¼ M n feg and e is unit element of M, and a strong epimorphism ðf ; gÞ of lattice-valued transformation monoids to the corresponding strong epimorphism ðf ; gÞ of lattice-valued transformation semigroups. The forgetful functor U q : qLtm ! qLts from the category qLtm of quasi-lattice-valued transformation monoids to the category qLts of quasi-lattice-valued transformation semigroups maps a quasi-lattice-valued transformation monoid ðQ ; M; qÞ to the underlying quasi-lattice-valued transformation semigroup ðQ ; S; qÞ where S ¼ M n feg and e is unit element of M, and a strong epimorphism ðf ; gÞ of quasi-lattice-valued transformation monoids to the corresponding strong epimorphism ðf ; gÞ of quasi-lattice-valued transformation semigroups. Other functors can be similarly defined. Theorem 3.1. The following diagram
Diagram 3.3 of the categories and functors is commutable. Proof. First, for all ðQ ; R; lÞ 2 Lfsm; ðEmq Extþ ÞðQ ; R; lÞ ¼ Emq ðQ ; Rþ ; lþ Þ ¼ ðQ ; R ; l Þ ¼ Ext ðQ ; R; lÞ by Diagrams 3.1 and 3.2. Second, for all ða; bÞ 2 LfsmðM1 ; M2 Þ; ðEmq Extþ Þða; bÞ ¼ Emq ðf a ; g b Þ ¼ ðf a ; g b Þ ¼ Ext ða; bÞ. So Emq Extþ ¼ Ext , i.e., Diagram 3.3 is commutable. h Theorem 3.2. The following diagram
Diagram 3.4 of the categories and functors is commutable. Proof. First, for all ðQ ; S; qÞ 2 qLts; ðEms Trþ ÞðQ ; S; qÞ ¼ Ems ðQ ; S ; q Þ ¼ ðQ ; M= ; q= Þ; where M ¼ S [ feg and e is the identity of M, and ðTr Emq ÞðQ ; S; qÞ ¼ Tr ðQ ; M; qÞ ¼ ðQ ; M= ; q= Þ.
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Second, for all ðf ; gÞ 2 qLts; Ems Tr þ Þðf ; gÞ ¼ Ems ðf ; g= Þ ¼ ðf ; g= Þ and ðTr Emq Þðf ; gÞ ¼ Tr ðf ; gÞ ¼ ðf ; g= Þ. Therefore Ems Trþ ¼ Tr Emq , i.e., Diagram 3.4 is commutable. h Theorem 3.3. The following diagram
Diagram 3.5 of the categories and functors is commutable. Proof. The proof is similar to Theorem 3.2.
h
Theorem 3.4. The following diagram
Diagram 3.6 of the categories and functors is commutable. Proof. The proof is similar to Theorem 3.1.
h
Theorem 3.5. Consider two functors Rþ : qLts ! Lfsm and Extþ : Lfsm ! qLts. The following conclusions are valid: (1) Extþ is a left adjoint to Rþ ; (2) There exist a natural transformations g : 1Lfsm ) Rþ Extþ and
e : Extþ Rþ ) 1qLts such that
ðRþ eÞ ðg Rþ Þ ¼ 1Rþ ; ðe Extþ Þ ðExtþ gÞ ¼ 1Extþ ; (3) There exist bijections
hSM : qLtsðExtþ ðMÞ; SÞ ffi LfsmðM; Rþ ðSÞÞ for any object S 2 ObðqLtsÞ; M 2 ObðLftmÞ and those bijections are natural both in S and in M; (4) Rþ is a right adjoint to Extþ .
Proof. We prove only (3) because (1) , (2) , (3) , (4). h Define
hSM : qLtsðExtþ ðMÞ; SÞ ffi LfsmðM; Rþ ðSÞÞ by hSM ððf ; gÞÞ ¼ ðaf ; bg Þ for all ðf ; gÞ 2 qLtsðExtþ ðMÞ; SÞ. For all ðf ; gÞ : S ! S 0 and ða; bÞ : M0 ! M, consider the following two diagrams
Diagram 3.7
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Diagram 3.8 Here ðf ; gÞ to be qLtsðExt ðMÞ; ðf ; gÞÞ a short, and ða; bÞ ¼ Lfsmðða; bÞ; Rþ ðSÞÞ. Since, for any ðp; qÞ 2 qLtsðExtþ ðMÞ; SÞ, there is þ
ðhS0 M ðf ; gÞ Þððp; qÞÞ ¼ hS0 M ðððf ; gÞ Þðp; qÞÞ ¼ hS0 M ððf ; gÞ ðp; qÞÞ ¼ hS0 M ððf p; g qÞÞ ¼ ðaf p ; bgq Þ ¼ ðaf ; bg Þ ðap ; bq Þ ¼ ðRþ ðf ; gÞ Þ ðap ; bq Þ ¼ ðRþ ðf ; gÞ Þ ðhSM ðp; qÞÞ ¼ ðRþ ðf ; gÞ hSM Þððp; qÞÞ; hence
hS0 M ððf ; gÞ Þ ¼ ðRþ ðf ; gÞ Þ hSM : Similarly,
hS0 M Extþ ðða; bÞÞ ¼ ðða; bÞ Þ hSM : Hence the two diagrams given above are commutable. Theorem 3.6. Consider the following two diagrams
Diagram 3.9
Diagram 3.10 Then the following conclusions are valid.
(1) (2) (3) (4) (5) (6)
Extþ is a left adjoint functor to Rþ ; Ext is a left adjoint functor to R ; Trþ is a left adjoint functor to Sþ ; Tr is a left adjoint functor to S ; Emq is a left adjoint functor to U q ; Ems is a left adjoint functor to U s .
Proof. The proof is similar to Theorem 3.5. h
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Acknowledgements Here please allow us to express our heart-felt thanks to Prof. Janusz Kacprzyk (Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland), and Prof. Ying-ming Liu (Sichuan University, Chengdu, P.R. China) for their responsible reading and helpful suggestions. References [1] W.G. Wee, On Generalizations of Adaptive Algorithm and Application of the Fuzzy Sets Concept to Pattern Classification, Ph, D. Thesis, Purdue University, June, 1967. [2] W.G. Wee, K.S. Fu, A formulation of fuzzy automata and its application as a model of learning systems, IEEE, Trans. Syst. Sci. Cybern., SSSC 5 (1969) 215–224. [3] E.S. Santos, Realization of fuzzy language by probabilistic, max-product and maximin automata, Inf. Sci. 8 (1975) 39–54. [4] E.S. Santos, Fuzzy automata and language, Inf. Sci. 10 (1976) 193–197. [5] E.S. Santos, Fuzzy automata and language, Inf. Sci. 10 (1976) 331–335. [6] E.S. Santos, Regular fuzzy expressions, in: M.M. Gupta, G.H. Saridis, B.R. Gaines (Eds.), Fuzzy automata and Decision Processes, Elsevier, North-Holland, 1977, pp. 169–175. [7] E.T. Lee, LA. Zadeh, Note on Fuzzy languages, Inf. Sci. 1 (1969) 421–435. [8] H.V. Kumbhojkar, S.R. Chaudhari, Coverings of products of fuzzy finite state machines, Fuzzy Sets Syst. 53 (1993) 203–216. [9] H.V. Kumbhojkar, S.R. Chaudhari, Fuzzy recognizers and recognizable sets, Fuzzy Sets Syst. 131 (2002) 381–391. [10] D.S. Malik, J.N. Mordeson, M.K. Sen, Semigroups of fuzzy finite state machines, in: P.P. Wang (Ed.), Advances in Fuzzy Technology, Bookswrite, Durham, North Carolima, 1995. [11] D.S. Malik, J.N. Mordeson, M.K. Sen, On subsystem of fuzzy finite state machines, Fuzzy Set Syst. 68 (1994) 83–92. [12] D.S. Malik, J.N. Mordeson, M.K. Sen, Submachines of fuzzy finite state machines, J. Fuzzy Math. 4 (1994) 781–792. [13] D.S. Malik, J.N. Mordeson, M.K. Sen, Admissible partitions of fuzzy finite state machine, Fuzzy Set Syst. 92 (1997) 95–102. [14] D.S. Malik, J.N. Mordeson, M.K. Sen, Products of fuzzy finite state machine, Int. J. Uncertain. Fuzz. Knowl.-Based Syst. 5 (6) (1997) 723–732. [15] J.N. Mordeson, D.S. Malik, Fuzzy Automata and Languages: Theory and Applications, Chapman Hall CRC, Boca Raton, London, New Yourk, Washington, DC, 2002. [16] G. Birkhoff, Lattice Theory, Amer. Math. Coll. Publ, third ed., vol. 25, AMS, New York, 1973. [17] J.E. Hopcroft, J.D. Ullman, Formal Languages and their Relation to Automata, Reading, Mass, Addison-Wesley, 1969.
Contents lists available at ScienceDirect
Information Sciences journal homepage: www.elsevier.com/locate/ins
Results on the use of category theory for the study of lattice-valued finite state machines q Jian-Gang Tang a,⇑, Mao-Kang Luo b, Juan Tang c a
College of Mathematics and Statistics, YiLi Normal University, Yining 835000, PR China College of Mathematics, Sichuan University, Chengdu 610064, PR China c College of Electronics and Information Engineering, Sichuan University, Chengdu 610064, PR China b
a r t i c l e
i n f o
Article history: Received 25 June 2006 Received in revised form 27 May 2014 Accepted 23 June 2014 Available online 31 July 2014 Keywords: Category of L-fuzzy finite state machines Category of L-fuzzy quasi-transformation semigroups Category of L-fuzzy quasi-transformation monoids Category of L-fuzzy transformation semigroups Category of L-fuzzy transformation monoids Functor
a b s t r a c t In this paper, we introduce the concepts of category of lattice-valued finite state machines, category of quasi-lattice-valued transformation semigroups, category of quasi-lattice-valued transformation monoids, category of lattice-valued transformation semigroups, category of lattice-valued transformation monoids and six pairs of functors, each pair of functors have the adjoint property, we discuss the relation between the categories with functors and give the adjoint theorems and relation theorems. 2014 Elsevier Inc. All rights reserved.
1. The concept of lattice-valued finite machines In 1967, Wee [1] firstly introduced the fuzzy finite automata, which generated many interesting explorations. Its important applications in learning systems, automatic control, pattern recognition and database, and so on were also studied by many researchers; the more details and the fuzzy finite automata were studied by Wee and Fu [2], Santos [3–6], Lee and Zadeh [7], Kumbhojkar and Chaudhari [8,9], Malik et al. [10–15], and so on; the authors can also refer to [16,17]. In this paper, we introduce the concept of lattice-valued finite machine extended from the concept of fuzzy finite machine in Wee [1]. For this purpose, we firstly introduce the concept of the Heyting algebra and the intuitionistic first-order logic as follow. Definition 1.1. A lattice L is called a Heyting algebra, if for any a; b 2 L, there exists an element ða ! bÞ 2 L such that
8c 2 L;
c 6 ða ! bÞ () c ^ a 6 b:
For any element a in a Heyting algebra L, especially denote :a ¼ ða ! 0Þ, call :a the negation of a. q
Supported by NSFC (Nos. 11161050, 31240020).
⇑ Corresponding author.
E-mail addresses: [email protected] (J.-G. Tang), [email protected] (M.-K. Luo). http://dx.doi.org/10.1016/j.ins.2014.06.027 0020-0255/ 2014 Elsevier Inc. All rights reserved.
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Any lattice ðL; 6Þ gives rise to a category, with the elements of L as objects, and with precisely one morphism from p to q if and only if p 6 q; in other words, the morphisms are pairs ðp; qÞ such that p 6 q, and the composition operation for L is uniquely determined by the transitivity of the order relation 6, and the mapping with order preserving between two complete lattices is the functor between two categories. Let L be a lattice. For every a 2 L; a ^ ðÞ : L ! L is an order-preserving mapping, by Definition 1.1, L is a Heyting algebra if and only if 8a 2 L, the functor
a ^ ðÞ : L ! L has a right adjoint
a ! ðÞ : L ! L: The following conclusion is often used in this paper, and its proof is easy: Theorem 1.1. Let L be a lattice, ? be a binary operation on L. Then ðL; !Þ is a Heyting algebra if and only if the following conditions hold: (1) (2) (3) (4)
a 6 b () a ! b ¼ 1, a ^ ða ! bÞ ¼ a ^ b, b ^ ða ! bÞ ¼ b, a ! ðb ^ aÞ ¼ ða ! bÞ ^ ða ! cÞ.
If L is a complete Heyting algebra, then 8a 2 L; a ! ðÞ : L ! L is an order-preserving mapping, and then, for any b 2 L, we have
a ! b ¼ _fx 2 Lja ^ x 6 bg: Syntax of the intuitionistic first-order logic is similar to the classical first-order logic. We have three primitive connective symbols : (negation), ^ (conjunction), and ? (implication) and a primitive quantifier " (universal quantifier). The connectives _ (disjunction), M (bi-implication) and the existential quantifier $ are defined in terms of :, ^, ?, and " in the usual way. In addition, let L be a Heyting algebra, 2 (membership) be a binary (primitive) predicate symbol. Then # (inclusion) and (equality) can be defined by 2 as usual. The semantics of the intuitionistic first-order logic is given by interpreting the connectives :, ^, and ? as the operations :ðÞ, ^, and ?, respectively, on L and by interpreting the quantifier " as the least upper bound in L. In addition, the truth value of the set-theoretical formula x 2 A is kx 2 Ak ¼ AðxÞ. It is worth indicating that in this paper the set A and its membership function are identified. In the intuitionistic first-order logic, 1 is the unique designated truth value. In other words, a formula u is valid if and only if its truth value kuk is 1. Definition 1.2. A lattice-valued finite state machine (or LFM for short) is defined as a triple M ¼ ðQ ; R; lÞ, where Q is a finite set of internal states; R is a finite set of symbols called the input alphabet; l is a lattice-valued subset of Q R Q , i.e., l is a function from Q R Q to L, where L is a Heyting algebra. If lðp; r; qÞ ¼ a with a 2 L, then p is said to be a previous state, r is said to be input, q is said to be a next state of p and a is said to be a membership degree or probable degree. The purpose of using lattice-valued sets in the finite automata is to interpret an imprecise character of changes in the internal state of the finite automata from one state to another state under an environmental change. It is more general to interpret this imprecision by using lattice-valued sets with a method of inferential flow with the multi-layer and the multi-branch which well be presented here. If lðp; r; qÞ ¼ a, where a 2 L, it can be interpreted that the lattice-valued finite state machine M is in a state p 2 Q and suppose it inputs r 2 R, it will cause the state to change on the membership degree a 2 L to a new state q 2 Q . And we say that the edge
begins at p, ends at q, and carries the label
ðr=aÞðsÞ ¼
a
if
r=a, where r=a can be regarded as a lattice-valued subset of R defined by
s ¼ r;
0 otherwise:
A (crisp) finite state machine ðQ ; R; dÞ, where d : Q R ! Q is a relation, can be considered as a lattice-valued finite state machine ðQ ; R; lÞ, if we define l : Q R Q ! L as follows:
lðp; r; qÞ ¼
1 if dðp; rÞ ¼ q; 0
otherwise:
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On the other hand, a lattice-valued finite state machine ðQ ; R; lÞ, when state machine ðQ ; R; dÞ, if we define
dðp; rÞ ¼ q if and only if
lðQ R Q Þ # f0; 1g, can be regarded as a finite
lðp; r; qÞ ¼ 1
However, in this situation d is not necessarily a partial function, but may happen to be a proper relation. In other words, ðQ ; R; dÞ may not be a deterministic finite state machine. A fuzzy finite state machine M ¼ ðQ ; R; lÞ [1] can also be regarded as a lattice-valued finite state machine, if we set L ¼ ½0; 1. 2. The category of lattice-valued finite state machines Definition 2.1. Let M1 ¼ ðQ 1 ; R1 ; l1 Þ and M2 ¼ ðQ 2 ; R2 ; l2 Þ be two lattice-valued finite state machines. Let a : Q 1 ! Q 2 and b : R1 ! R2 be mappings. A pair (a, b) is called a strong homomorphism, written ða; bÞ : M1 ! M2 , if
l2 ðaðqÞ; bðxÞ; aðpÞÞ ¼ _fl1 ðq; x; tÞjt 2 Q 1 ; aðtÞ ¼ aðpÞg; 8q; p 2 Q 1 and 8x 2 R1 . A strong homomorphism ða; bÞ : M1 ! M2 is called a strong monomorphism (strong epimorphism, or strong isomorphism), if both the mappings a and b are injective (surjective, or bijective).
Definition 2.2. The category of lattice-valued finite state machines (or Lfsm for short) is defined as follows: object: lattice-valued finite state machine M ¼ ðQ ; R; lÞ; morphism: strong epimorphisms ða; bÞ : M1 ! M2 . Let M ¼ ðQ ; R; lÞ be a lattice-valued finite state machine where Q and R are non-empty finite sets and l is a lattice-valued subset of Q R Q ; R is called the alphabet, an element of R is called symbol, Rþ is a set of words over R, and R is Rþ adding the empty word e, i.e. R ¼ Rþ [ feg. It is obvious that R is a free monoid on R and Rþ is a free semigroup on R, respectively, under usual juxtaposition of words. A lattice-valued set l : Q R Q ! L can induce two functions lþ : Q Rþ Q ! L and l : Q R Q ! L defined by
lþ ðp; r; qÞ ¼ lðp; r; qÞ; lþ ðp; xr; qÞ ¼ _flþ ðp; x; rÞ ^ lðr; r; qÞjr 2 Q g; for all p; q 2 Q ; x 2 Rþ ; r 2 R. And l satisfies l jRþ ¼ lþ with
l ðp; e; qÞ ¼
1 if p ¼ q 0
if p – q:
Theorem 2.1. Let M ¼ ðQ ; R; lÞ be a lattice-valued finite state machine. Then
lþ ðp; xy; qÞ ¼
_ flþ ðp; x; rÞ ^ lþ ðr; y; qÞÞjr 2 Qg;
for all p; q 2 Q and x; y 2 Rþ . And
l is the same.
Proof. Let x ¼ a1 ; . . . ; an and y ¼ b1 ; . . . ; bm with a1 ; . . . ; an ; b1 ; . . . ; bm 2 R. Then
_
flþ ðp; x; rÞ ^ lþ ðr; y; qÞÞjr 2 Q g ¼
_
flþ ðp; a1 ; . . . ; an ; rÞ ^ lþ ðr; b1 ; . . . ; bm ; qÞjr 2 Q g _ n_ ¼ flðp; a1 ; q1 Þ ^ ^ lðqn1 ; an ; rÞjq1 ; . . . ; qn1 2 Qg ^
_
o flðr; b1 ; qn Þ ^ ^ lðqnþm1 ; bm ; qÞjqn ; . . . ; qnþm1 2 Q g r 2 Q
_
flðp; a1 ; q1 Þ ^ ^ lðqn1 ; an ; rÞ ^ lðr; b1 ; qn Þ ^ ^lðqnþm ; bm ; qÞÞjq1 ; . . . ; qnþm1 ; r 2 Q
¼
¼ lþ ðp; a1 ; . . . ; an b1 ; . . . ; bm ; qÞ ¼ lþ ðp; xy; qÞ:
þ Let b be a function from a set R1 to a set R2 . Then, bþ is the homomorphism of semigroups from Rþ 1 to R2 defined by
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bþ ðx1 ; . . . ; xn Þ ¼ bðx1 Þ; . . . ; bðxn Þ; for any xi 2 R1 ; i ¼ 1; 2; . . . ; n. And b is the homomorphism of monoids from R1 to R2 defined by
b ðx1 ; . . . ; xn Þ ¼ bðx1 Þ; . . . ; bðxn Þ; for any xi 2 R1 ; i ¼ 1; 2; . . . ; n. Then bþ (b ) is called an extended homomorphism of semigroups (monoids). Theorem 2.2. Let M1 ¼ ðQ 1 ; R1 ; l1 Þ and M2 ¼ ðQ 2 ; R2 ; l2 Þ be lattice-valued finite state machines, ða; bÞ : M1 ! M2 a strong homomorphism. Then
lþ2 ðaðqÞ; bþ ðxÞ; aðpÞÞ ¼ _flþ1 ðq; x; tÞjt 2 Q 1 ; aðtÞ ¼ aðpÞg; for any q; p 2 Q 1 and 8x 2 Rþ 1 . And
l2 ðaðqÞ; b ðxÞ; aðpÞÞ ¼ _fl1 ðq; x; tÞjt 2 Q 1 ; aðtÞ ¼ aðpÞg; for any q; p 2 Q 1 and 8x 2 R1 . Proof. The proof is straightforward. h Definition 2.3. A quasi-lattice-valued transformation semigroup is a triple ðQ ; S; qÞ consisting of a finite set Q, a finite semigroup S and a lattice-valued subset of Q S Q q, i.e., q is a function from Q S Q to L, and W qðp; uv ; qÞ ¼ fqðp; u; rÞ ^ qðr; v ; qÞÞjr 2 Q g for all p; q 2 Q and u; v 2 S. A quasi-lattice-valued transformation semigroup S ¼ ðQ ; S; qÞ is regarded as a lattice-valued finite state machine ðQ ; R; lq Þ by setting R to S and lq ¼ q, we denote it as FSMðSÞ. Definition 2.4. A quasi-lattice-valued transformation semigroup S ¼ ðQ ; S; qÞ is called a quasi-lattice-valued transformation monoid if S is a monoid with an identity e such that the following condition holds:
qðp; e; qÞ ¼
1 if p ¼ q; 0 if p – q:
A quasi-lattice-valued transformation monoid S ¼ ðQ ; S; qÞ can also be regarded as a lattice-valued finite state machine ðQ ; R; lq Þ by putting R to be S and lq ¼ q, we also denote it as FSMðSÞ. Definition 2.5. Let S 1 ¼ ðQ 1 ; S1 ; q1 Þ and S 2 ¼ ðQ 2 ; S2 ; q2 Þ be two quasi-lattice-valued transformation semigroups. Let f : Q 1 ! Q 2 be a mapping and g : S1 ! S2 a homomorphism of semigroups. A pair ðf ; gÞ is called a strong homomorphism, written ðf ; gÞ : S 1 ! S 2 , if
q2 ðf ðqÞ; gðxÞ; f ðpÞÞ ¼ _fq1 ðq; x; tÞjt 2 Q 1 ; f ðtÞ ¼ f ðpÞg; 8q; p 2 Q 1 and 8x 2 S1 . A strong homomorphism ðf ; gÞ : S 1 ! S 2 is called a strong monomorphism (strong epimorphism, or strong isomorphism), if both the mappings a and b are injectives (surjectives, or bijectives). Definition 2.6. The category of quasi-lattice-valued transformation semigroups (or qLts for short) is defined as follows: objects: quasi-lattice-valued transformation semigroup S ¼ ðQ ; S; qÞ; morphism: strong epimorphisms ðf ; gÞ : S 1 ! S 2 . We can also define the category qLtm of quasi-lattice-valued transformation monoids with quasi-lattice-valued transformation monoid as object and the strong epimorphism of quasi-lattice-valued transformation monoids as morphism. A lattice-valued finite state machine M ¼ ðQ ; R; lÞ naturally introduces a quasi-lattice-valued transformation semigroup ðQ ; Rþ ; ql Þ, where ql is defined by
ql ðp; x; qÞ ¼ lþ ðp; x; qÞ: We call ðQ ; Rþ ; ql Þ a quasi-lattice-valued transformation semigroup introduced by M and denote it as qFTSðMÞ, i.e. qFTSðMÞ ¼ ðQ ; Rþ ; ql Þ. Theorem 2.3. Let M ¼ ðQ ; R; lÞ be a lattice-valued finite state machine, then ðQ ; Rþ ; ql Þ given by M is a quasi-lattice-valued transformation semigroup. It is clear that ðQ ; Rþ ; ql Þ is a quasi-lattice-valued transformation semigroup, and then, we can additionally define
ql ðp; e; qÞ ¼
1 if p ¼ q; 0
if p – q:
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Then ðQ ; R ; ql Þ given by M is a quasi-lattice-valued transformation monoid and denote it as qFTMðMÞ, i.e. qFTMðMÞ ¼ ðQ ; R ; ql Þ . Theorem 2.4. Let M ¼ ðQ ; R; lÞ be a lattice-valued finite state machine, then ðQ ; R ; ql Þ introduced by M is a quasi-latticevalued transformation monoid. Theorem 2.5. Let M1 ¼ ðQ 1 ; R1 ; l1 Þ and M2 ¼ ðQ 2 ; R2 ; l2 Þ be two lattice-valued finite state machines and ða; bÞ : M2 ! M2 a strong epimorphism of lattice-valued finite state machines. Then there exists a strong epimorphism of quasi-lattice-valued transformation semigroups (monoids)
ðf a ; g b Þ : qFTSðM1 Þ ! qFTSðM2 Þ; ððf a ; g b Þ : qFTMðM1 Þ ! qFTMðM2 ÞÞ: Proof. The proof is straightforward. h Theorem 2.6. Let S 1 ¼ ðQ 1 ; S1 ; q1 Þ and S 2 ¼ ðQ 2 ; S2 ; q2 Þ be two quasi-lattice-valued transformation semigroups (quasi-latticevalued transformation monoids) and ðf ; gÞ : S 1 ! S 2 a strong epimorphism of quasi-lattice-valued transformation semigroups (quasi-lattice-valued transformation monoids). Then there exists a strong epimorphism of lattice-valued finite state machines
ðaf ; bg Þ : FSMðS 1 Þ ! FSMðS 2 Þ: Proof. Let FSMðS 1 Þ ¼ ðQ 1 ; R1 ; lq1 Þ; FSMðS 2 Þ ¼ ðQ 2 ; R2 ; lq2 Þ and lq1 ¼ q1 ; lq2 ¼ q2 , respectively. Define af : Q 1 ! Q 2 by af ¼ f and bg : R1 ! R2 by bg ¼ g. Then
lq2 ðaf ðqÞ; bg ðxÞ; af ðpÞÞ ¼ q2 ðf ðqÞ; gðxÞ; f ðpÞÞ ¼ _fq1 ðq; x; tÞjt 2 Q 1 ; f ðtÞ ¼ f ðpÞg ¼ _flq1 ðq; x; tÞjt 2 Q 1 ; af ðtÞ ¼ af ðpÞg 8q; p 2 Q 1 and 8x 2 R1 . Hence there exists a strong epimorphism of lattice-valued finite state machines ðaf ; bg Þ : FSMðS 1 Þ ! FSMðS 2 Þ: For a quasi-lattice-valued finite semigroup S = ðQ ; S; qÞ, we can define a relation on S by x y if and only if
qðp; x; qÞ ¼ qðp; y; qÞ for all p; q 2 Q . h Theorem 2.7. Let S ¼ ðQ ; S; qÞ be a quasi-lattice-valued finite semigroup. Then is a congruence relation on S. Proof. Clearly, is an equivalence relation on S, let z2S and x y, then for all W W p; q 2 Q ; qðp; xz; qÞ ¼ fqðp; x; rÞ ^ qðr; z; qÞjr 2 Q g ¼ fqðp; y; rÞ ^ qðr; z; qÞjr 2 Q g ¼ qðp; yz; qÞ by Theorem 3.1, hence xz yz. Similarly, zx zy. Thus is a congruence relation on S. Given a quasi-lattice-valued finite semigroup S ¼ ðQ ; S; qÞ, let ½x ¼ fy 2 Rþ jx yg, where x 2 S and let S= ¼ f½xjx 2 Sg be denoted by Q ðSÞ. h Theorem 2.8. Let S ¼ ðQ ; S; qÞ be a lattice-valued finite semigroup. Then Q ðSÞ is a semigroup, the binary operation on Q ðSÞ is defined by ½x½y ¼ ½xy. Given a quasi-lattice-valued finite monoid S ¼ ðQ ; M; qÞ. We also can define a congruence relation on M, and then, get a monoid M= ¼ f½xjx 2 Mg which is denoted by Q ðSÞ, the binary operation on Q ðSÞ is defined by ½x½y ¼ ½xy. Definition 2.7. A lattice-valued transformation semigroup is a triple ðQ ; S; qÞ consisting of a finite set Q, a finite semigroup S and a lattice-valued subset of Q S Q q, i.e., q is a function from Q S Q to L, and the following conditions hold: W (1) qðp; uv ; qÞ ¼ fqðp; u; rÞ ^ qðr; v ; qÞÞjr 2 Q g for all p; q 2 Q and u; v 2 S; (2) For u; v 2 S, if qðp; u; qÞ ¼ qðp; v ; qÞ for all p; q 2 Q , then u ¼ v . A lattice-valued transformation semigroup S ¼ ðQ ; S; qÞ is regarded as a quasi-lattice-valued finite semigroup immediately. A lattice-valued transformation semigroup S ¼ ðQ ; S; qÞ is regarded as a lattice-valued finite state machine ðQ ; R; lq Þ by setting R to be S and lq ¼ q, we denote it as FSMðSÞ. Definition 2.8. A lattice-valued transformation semigroup S ¼ ðQ ; S; qÞ is called a lattice-valued transformation monoid if S is a monoid with an identity e such that the following condition holds:
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qðp; e; qÞ ¼
1 if p ¼ q; 0 if p – q:
A lattice-valued transformation monoid S ¼ ðQ ; S; qÞ can be regarded as a quasi-lattice-valued finite monoid immediately. A lattice-valued transformation monoid S ¼ ðQ ; S; qÞ can also be regarded as a lattice-valued finite state machine ðQ ; S; lq Þ by setting R to be M and lq ¼ q, we also denote it as FSMðSÞ. Definition 2.9. Let S 1 ¼ ðQ 1 ; S1 ; q1 Þ and S 2 ¼ ðQ 2 ; S2 ; q2 Þ be two lattice-valued transformation semigroups. Let f : Q 1 ! Q 2 be a mapping and g : S1 ! S2 a homomorphism of semigroups. A pair ðf ; gÞ is called a strong homomorphism, written as ðf ; gÞ : S 1 ! S 2 , if
q2 ðf ðqÞ; gðxÞ; f ðpÞÞ ¼ _fq1 ðq; x; tÞjt 2 Q 1 ; f ðtÞ ¼ f ðpÞg; 8q; p 2 Q 1 and 8x 2 S1 . A strong homomorphism ðf ; gÞ : S 1 ! S 2 is called a strong monomorphism (strong epimorphism, or strong isomorphism), if both the mappings a and b are injective (surjective, or bijective). Definition 2.10. The category of lattice-valued transformation semigroups (or Lts for short) is defined as follows objects: lattice-valued transformation semigroup S ¼ ðQ ; S; qÞ; morphism: strong epimorphisms ðf ; gÞ : S 1 ! S 2 . We can also define the strong epimorphism of lattice-valued transformation monoids and the category Ltm of lattice-valued transformation monoids. Now we consider a lattice-valued transformation semigroup and a lattice-valued transformation monoid introduced by a quasi-lattice-valued transformation semigroup and a lattice-valued transformation monoid, respectively. A quasi-lattice-valued transformation semigroup S ¼ ðQ ; S; qÞ naturally introduces a lattice-valued transformation semigroup ðQ ; S= ; q= Þ, where q= is defined by
q= ðp; ½x; qÞ ¼ qðp; x; qÞ for all ½x 2 S= . We call ðQ ; S= ; q= Þ as a lattice-valued transformation semigroup introduced by S and denote it as FTSðSÞ, i.e. FTSðSÞ ¼ ðQ ; S= ; q= Þ. A quasi-lattice-valued transformation monoid S ¼ ðQ ; M; qÞ can also naturally introduces a lattice-valued transformation monoid ðQ ; M= ; q= Þ, it denoted by FTMðSÞ, i.e. FTMðSÞ ¼ ðQ ; S= ; q= Þ. Theorem 2.9. Let S 1 ¼ ðQ 1 ; S1 ; q1 Þ and S 2 ¼ ðQ 2 ; S2 ; q2 Þ be two quasi-lattice-valued transformation semigroups and ðf ; gÞ : S 2 ! S 2 a strong epimorphism of quasi-lattice-valued transformation semigroups. Then there exists a strong epimorphism of lattice-valued transformation semigroups
ðf ; g= Þ : FTSðS 1 Þ ! FTSðS 2 Þ: Proof. Let S1 = ¼ Q ðS 1 Þ; S2 = ¼ Q ðS 2 Þ and suppose s 2 S1 = , then there exists a 2 S1 such that s ¼ ½a, the -equivalence class containing a. Suppose a ¼ r1 ; . . . ; rn ; ri 2 S1 , we define g= ðsÞ ¼ ½gðaÞ0 where ½gðaÞ0 is the 0 -equivalence class containing gðaÞ ¼ gðr1 Þ; . . . ; gðrn Þ 2 S2 . (Note that is induced by S 1 and 0 is induced by S 2 .) We must firstly establish that g= : S1 = ! S2 =0 is well-defined. Suppose that s ¼ ½b, where b ¼ s1 ; . . . ; sm with sj 2 S1 . Now for any q; p 2 Q 1 ; q1 ðq; a; pÞ ¼ q1 ðq; b; pÞ, let q0 ; p0 2 Q 2 , there exists q; p 2 Q 1 such that q0 ¼ f ðqÞ and p0 ¼ f ðpÞ, respectively. Then
q2 ðq0 ; gðaÞ; p0 Þ ¼ q2 ðf ðqÞ; gðaÞ; f ðpÞÞ ¼ _fq1 ðq; a; tÞjt 2 Q 1 ; f ðtÞ ¼ f ðpÞg ¼ _fq1 ðq; b; tÞjt 2 Q 1 ; f ðtÞ ¼ f ðpÞg ¼ q2 ðf ðqÞ; gðbÞ; f ðpÞÞ ¼ q2 ðq0 ; gðbÞ; p0 Þ: Thus, gðaÞ0 gðbÞ and g= is well-defined. Now, let q; p 2 Q ; s ¼ ½a 2 S1 = , then
q2 = ðf ðqÞ; g= ðsÞ; f ðpÞÞ ¼ q2 = ðf ðqÞ; g= ð½aÞ; f ðpÞÞ ¼ q2 = ðf ðqÞ; ½gðaÞ0 ; f ðpÞÞ ¼ q2 ðf ðqÞ; gðaÞ; f ðpÞÞ ¼ _fq1 ðq; a; tÞjt 2 Q 1 ; f ðtÞ ¼ f ðpÞg ¼ _fq1 = ðq; ½a; tÞjt 2 Q 1 ; f ðtÞ ¼ f ðpÞg ¼ _fq1 = ðq; s; tÞjt 2 Q 1 ; f ðtÞ ¼ f ðpÞg
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where s ¼ ½a and a 2 S1 . So ðf ; g= Þ : FTSðS 1 Þ ! FTSðS 2 Þ is a strong epimorphism of lattice-valued transformation semigroups. Theorem 2.10. Let S 1 ¼ ðQ 1 ; S1 ; q1 Þ and S 2 ¼ ðQ 2 ; S2 ; q2 Þ be two quasi-lattice-valued transformation monoids and ðf ; gÞ : S 2 ! S 2 a strong epimorphism of quasi-lattice-valued transformation monoids. Then there exists a strong epimorphism of lattice-valued transformation monoids
ðf ; g= Þ : FTMðS 1 Þ ! FTMðS 2 Þ: Proof. The proof is similar to Theorem 3.9.
h
Theorem 2.11. Let S 1 ¼ ðQ 1 ; S1 ; q1 Þ and S 2 ¼ ðQ 2 ; S2 ; q2 Þ be two lattice-valued transformation semigroups and ðf ; gÞ : S 1 ! S 2 a strong epimorphism of lattice-valued transformation semigroups. Then there exists a strong epimorphism of quasi-lattice-valued transformation semigroups (quasi-lattice-valued transformation monoids).
ðf ; gÞ : S 1 ! S 2 : Proof. The proof is straightforward. h
3. Main results Category theory is a very general tool that can define, analyze, compare, etc. at a macro (very general) level different concepts and properties. Automata, including the lattice-valued finite state machines, are very general tools that can be formalized in many different ways, applied in many different fields, serve for so different purposes, etc. Hence, the use of categorytheoretical tools for the analysis and synthesis of automata, including the lattice-valued finite state machines, makes much sense. Our main results obtained in this paper can be briefly summarized as: From a macro (very general) point of view, a latticevalued finite state machines is equivalent (in terms of its behavior) to a lattice-valued transformation semigroup, and we can use well-known results from the theory of semigroups to study the lattice-valued finite state machines. This is a very important result because the theory of semigroups is a general and powerful apparatus, and it may give us many tools. Consider the diagram of the categories and the functors.
Diagram 3.1 In Diagram 3.1, Lfsm is the category of lattice-valued finite state machines, qLts is the category of quasi-lattice-valued transformation semigroups, qLtm is the category of quasi-lattice-valued transformation monoids, Lts is the category of lattice-valued transformation semigroups, and Ltm is the category of lattice-valued transformation monoids. Extþ and Ext are the extending functors, Tr þ and Tr are the transforming functors, Emp and Ems are the embedding functors. These functors are defined as follows: The extending functor Extþ : Lsm ! qLts from the category Lfsm of lattice-valued finite state machines to the category qLts of quasi-lattice-valued transformation semigroups is defined as follows: For all M ¼ ðQ ; R; lÞ 2 ObðLfsmÞ; Extþ ðMÞ ¼ qFTSðMÞ ¼ ðQ ; Rþ ; lþ Þ 2 ObðqLtsÞ is a quasi-lattice-valued transformation semigroup introduced by M. For all strong epimorphism of lattice-valued finite state machines ða; bÞ : ðQ 1 ; R1 ; l1 Þ ! ðQ 2 ; R2 ; l2 Þ; Extþ ðða; bÞÞ ¼ þ þ þ ðf a ; g b Þ : ðQ 1 ; Rþ 1 ; l1 Þ ! ðQ 2 ; R2 ; l2 Þ is a strong epimorphism of quasi-lattice-valued transformation semigroups according to Theorem 2.5. The extending functor Ext : Lfsm ! qLtm from the category Lfsm of lattice-valued finite state machines to the category qLtm of quasi-lattice-valued transformation monoids is defined as follows:
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For all M ¼ ðQ ; R; lÞ 2 ObðLfsmÞ, Ext ðMÞ ¼ qFTMðMÞ ¼ ðQ ; R ; l Þ 2 ObðqLtmÞ is a quasi-lattice-valued transformation monoid introduced by M. For all strong epimorphism of lattice-valued finite state machines ða; bÞ : ðQ 1 ; R1 ; l1 Þ ! ðQ 2 ; R2 ; l2 Þ; Ext ðða; bÞÞ ¼ ðf a ; g b Þ : ðQ 1 ; R1 ; l1 Þ ! ðQ 2 ; R2 ; l2 Þ is a strong epimorphism of quasi-lattice-valued transformation monoids according to Theorem 2.5. Other functors can be similarly defined. Consider the diagram of categories and the functors.
Diagram 3.2 U s and U p are the forgetful functors, S and Sþ are the shifting functors, R and Rþ are the restricting functors. These functors are defined as follows: The forgetful functor U s : Ltm ! Lts from the category Ltm of lattice-valued transformation monoids to the category Lts of lattice-valued transformation semigroups maps a lattice-valued transformation monoid ðQ ; M; qÞ to the underlying lattice-valued transformation semigroup ðQ ; S; qÞ where S ¼ M n feg and e is unit element of M, and a strong epimorphism ðf ; gÞ of lattice-valued transformation monoids to the corresponding strong epimorphism ðf ; gÞ of lattice-valued transformation semigroups. The forgetful functor U q : qLtm ! qLts from the category qLtm of quasi-lattice-valued transformation monoids to the category qLts of quasi-lattice-valued transformation semigroups maps a quasi-lattice-valued transformation monoid ðQ ; M; qÞ to the underlying quasi-lattice-valued transformation semigroup ðQ ; S; qÞ where S ¼ M n feg and e is unit element of M, and a strong epimorphism ðf ; gÞ of quasi-lattice-valued transformation monoids to the corresponding strong epimorphism ðf ; gÞ of quasi-lattice-valued transformation semigroups. Other functors can be similarly defined. Theorem 3.1. The following diagram
Diagram 3.3 of the categories and functors is commutable. Proof. First, for all ðQ ; R; lÞ 2 Lfsm; ðEmq Extþ ÞðQ ; R; lÞ ¼ Emq ðQ ; Rþ ; lþ Þ ¼ ðQ ; R ; l Þ ¼ Ext ðQ ; R; lÞ by Diagrams 3.1 and 3.2. Second, for all ða; bÞ 2 LfsmðM1 ; M2 Þ; ðEmq Extþ Þða; bÞ ¼ Emq ðf a ; g b Þ ¼ ðf a ; g b Þ ¼ Ext ða; bÞ. So Emq Extþ ¼ Ext , i.e., Diagram 3.3 is commutable. h Theorem 3.2. The following diagram
Diagram 3.4 of the categories and functors is commutable. Proof. First, for all ðQ ; S; qÞ 2 qLts; ðEms Trþ ÞðQ ; S; qÞ ¼ Ems ðQ ; S ; q Þ ¼ ðQ ; M= ; q= Þ; where M ¼ S [ feg and e is the identity of M, and ðTr Emq ÞðQ ; S; qÞ ¼ Tr ðQ ; M; qÞ ¼ ðQ ; M= ; q= Þ.
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Second, for all ðf ; gÞ 2 qLts; Ems Tr þ Þðf ; gÞ ¼ Ems ðf ; g= Þ ¼ ðf ; g= Þ and ðTr Emq Þðf ; gÞ ¼ Tr ðf ; gÞ ¼ ðf ; g= Þ. Therefore Ems Trþ ¼ Tr Emq , i.e., Diagram 3.4 is commutable. h Theorem 3.3. The following diagram
Diagram 3.5 of the categories and functors is commutable. Proof. The proof is similar to Theorem 3.2.
h
Theorem 3.4. The following diagram
Diagram 3.6 of the categories and functors is commutable. Proof. The proof is similar to Theorem 3.1.
h
Theorem 3.5. Consider two functors Rþ : qLts ! Lfsm and Extþ : Lfsm ! qLts. The following conclusions are valid: (1) Extþ is a left adjoint to Rþ ; (2) There exist a natural transformations g : 1Lfsm ) Rþ Extþ and
e : Extþ Rþ ) 1qLts such that
ðRþ eÞ ðg Rþ Þ ¼ 1Rþ ; ðe Extþ Þ ðExtþ gÞ ¼ 1Extþ ; (3) There exist bijections
hSM : qLtsðExtþ ðMÞ; SÞ ffi LfsmðM; Rþ ðSÞÞ for any object S 2 ObðqLtsÞ; M 2 ObðLftmÞ and those bijections are natural both in S and in M; (4) Rþ is a right adjoint to Extþ .
Proof. We prove only (3) because (1) , (2) , (3) , (4). h Define
hSM : qLtsðExtþ ðMÞ; SÞ ffi LfsmðM; Rþ ðSÞÞ by hSM ððf ; gÞÞ ¼ ðaf ; bg Þ for all ðf ; gÞ 2 qLtsðExtþ ðMÞ; SÞ. For all ðf ; gÞ : S ! S 0 and ða; bÞ : M0 ! M, consider the following two diagrams
Diagram 3.7
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Diagram 3.8 Here ðf ; gÞ to be qLtsðExt ðMÞ; ðf ; gÞÞ a short, and ða; bÞ ¼ Lfsmðða; bÞ; Rþ ðSÞÞ. Since, for any ðp; qÞ 2 qLtsðExtþ ðMÞ; SÞ, there is þ
ðhS0 M ðf ; gÞ Þððp; qÞÞ ¼ hS0 M ðððf ; gÞ Þðp; qÞÞ ¼ hS0 M ððf ; gÞ ðp; qÞÞ ¼ hS0 M ððf p; g qÞÞ ¼ ðaf p ; bgq Þ ¼ ðaf ; bg Þ ðap ; bq Þ ¼ ðRþ ðf ; gÞ Þ ðap ; bq Þ ¼ ðRþ ðf ; gÞ Þ ðhSM ðp; qÞÞ ¼ ðRþ ðf ; gÞ hSM Þððp; qÞÞ; hence
hS0 M ððf ; gÞ Þ ¼ ðRþ ðf ; gÞ Þ hSM : Similarly,
hS0 M Extþ ðða; bÞÞ ¼ ðða; bÞ Þ hSM : Hence the two diagrams given above are commutable. Theorem 3.6. Consider the following two diagrams
Diagram 3.9
Diagram 3.10 Then the following conclusions are valid.
(1) (2) (3) (4) (5) (6)
Extþ is a left adjoint functor to Rþ ; Ext is a left adjoint functor to R ; Trþ is a left adjoint functor to Sþ ; Tr is a left adjoint functor to S ; Emq is a left adjoint functor to U q ; Ems is a left adjoint functor to U s .
Proof. The proof is similar to Theorem 3.5. h
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Acknowledgements Here please allow us to express our heart-felt thanks to Prof. Janusz Kacprzyk (Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland), and Prof. Ying-ming Liu (Sichuan University, Chengdu, P.R. China) for their responsible reading and helpful suggestions. References [1] W.G. Wee, On Generalizations of Adaptive Algorithm and Application of the Fuzzy Sets Concept to Pattern Classification, Ph, D. Thesis, Purdue University, June, 1967. [2] W.G. Wee, K.S. Fu, A formulation of fuzzy automata and its application as a model of learning systems, IEEE, Trans. Syst. Sci. Cybern., SSSC 5 (1969) 215–224. [3] E.S. Santos, Realization of fuzzy language by probabilistic, max-product and maximin automata, Inf. Sci. 8 (1975) 39–54. [4] E.S. Santos, Fuzzy automata and language, Inf. Sci. 10 (1976) 193–197. [5] E.S. Santos, Fuzzy automata and language, Inf. Sci. 10 (1976) 331–335. [6] E.S. Santos, Regular fuzzy expressions, in: M.M. Gupta, G.H. Saridis, B.R. Gaines (Eds.), Fuzzy automata and Decision Processes, Elsevier, North-Holland, 1977, pp. 169–175. [7] E.T. Lee, LA. Zadeh, Note on Fuzzy languages, Inf. Sci. 1 (1969) 421–435. [8] H.V. Kumbhojkar, S.R. Chaudhari, Coverings of products of fuzzy finite state machines, Fuzzy Sets Syst. 53 (1993) 203–216. [9] H.V. Kumbhojkar, S.R. Chaudhari, Fuzzy recognizers and recognizable sets, Fuzzy Sets Syst. 131 (2002) 381–391. [10] D.S. Malik, J.N. Mordeson, M.K. Sen, Semigroups of fuzzy finite state machines, in: P.P. Wang (Ed.), Advances in Fuzzy Technology, Bookswrite, Durham, North Carolima, 1995. [11] D.S. Malik, J.N. Mordeson, M.K. Sen, On subsystem of fuzzy finite state machines, Fuzzy Set Syst. 68 (1994) 83–92. [12] D.S. Malik, J.N. Mordeson, M.K. Sen, Submachines of fuzzy finite state machines, J. Fuzzy Math. 4 (1994) 781–792. [13] D.S. Malik, J.N. Mordeson, M.K. Sen, Admissible partitions of fuzzy finite state machine, Fuzzy Set Syst. 92 (1997) 95–102. [14] D.S. Malik, J.N. Mordeson, M.K. Sen, Products of fuzzy finite state machine, Int. J. Uncertain. Fuzz. Knowl.-Based Syst. 5 (6) (1997) 723–732. [15] J.N. Mordeson, D.S. Malik, Fuzzy Automata and Languages: Theory and Applications, Chapman Hall CRC, Boca Raton, London, New Yourk, Washington, DC, 2002. [16] G. Birkhoff, Lattice Theory, Amer. Math. Coll. Publ, third ed., vol. 25, AMS, New York, 1973. [17] J.E. Hopcroft, J.D. Ullman, Formal Languages and their Relation to Automata, Reading, Mass, Addison-Wesley, 1969.