Recently, two types of simulations (forward and backward simulations) and four types of bisimulat... more Recently, two types of simulations (forward and backward simulations) and four types of bisimulations (forward, backward, forward-backward, and backward-forward bisimulations) between fuzzy automata have been introduced. If there is at least one simulation/bisimulation of some of these types between the given fuzzy automata, it has been proved that there is the greatest simulation/bisimulation of this kind. In the present paper, for any of the above-mentioned types of simulations/bisimulations we provide an effective algorithm for deciding whether there is a simulation/bisimulation of this type between the given fuzzy automata, and for computing the greatest one, whenever it exists. The algorithms are based on the method developed in [J. Ignjatović, M.Ćirić, S. Bogdanović, On the greatest solutions to certain systems of fuzzy relation inequalities and equations, Fuzzy Sets and Systems 161 (2010) 3081-3113], which comes down to the computing of the greatest post-fixed point, contained in a given fuzzy relation, of an isotone function on the lattice of fuzzy relations.
The Existence of Generalized Inverses of Fuzzy Matrices
Studies in Computational Intelligence, 2018
In this paper we show that every fuzzy matrix with entries in a complete residuated lattice posse... more In this paper we show that every fuzzy matrix with entries in a complete residuated lattice possess the generalized inverses of certain types, and in particular, it possess the greatest generalized inverses of these types. We also provide an iterative method for computing these greatest generalized inverses, which terminates in a finite number of steps, for example, for all fuzzy matrices with entries in a Heyting algebra. For other types of generalized inverses we determine criteria for the existence, given in terms of solvability of particular systems of linear matrix equations. When these criteria are met, we prove that there is the greatest generalized inverse of the given type and provide a direct method for its computing.
In this paper we provide procedures for testing the existence of various types of generalized inv... more In this paper we provide procedures for testing the existence of various types of generalized inverses in involutive residuated semigroups and involutive quantales defined by the Moore-Penrose equations, and for computing the extreme inverses whenever they exist. We also determine certain instances when a? = a*, whenever a? exists. The obtained results can be applied to a wide class of quantales of fuzzy relations and fuzzy matrices, as well as to Gelfand quantales.
In this paper we provide improvements of determinization methods, based on factorization of fuzzy... more In this paper we provide improvements of determinization methods, based on factorization of fuzzy states, for fuzzy finite automata that accept fuzzy languages of infinite range. The improvements are based on the usage of the fuzzy relational calculus, namely, on the usage of the right invariant fuzzy quasi-orders. Our algorithms perform better in the sense that they produce smaller automata, while require the same computation time. In addition, they can produce finite deterministic automata in cases when previous algorithms result in infinite deterministic automata. We show that the weak representable-cycles property is necessary and sufficient condition for determinization of a fuzzy automaton via a maximal factorization of fuzzy states. This condition is more general than the representable-cycles property previously determined as the necessary and sufficient condition for determinization of a fuzzy automaton via a maximal factorization of fuzzy states.
In this paper we consider different types of ranks of fuzzy matrices over residuated lattices. We... more In this paper we consider different types of ranks of fuzzy matrices over residuated lattices. We investigate relations between ranks and prove that row rank, column rank and Schein rank of idempotent fuzzy matrices are equal. In particular, ranks and corresponding decompositions of fuzzy matrices representing fuzzy quasi-orders are studied in detail. We show that fuzzy matrix decomposition by ranks can be used in the state reduction of fuzzy automata. Moreover, we prove that using rank decomposition of fuzzy matrices improves results of any state reduction method based on merging indistinguishable states of fuzzy automata.
In this paper we study several types of systems of fuzzy relation equations and inequalities comp... more In this paper we study several types of systems of fuzzy relation equations and inequalities composed of a given family of fuzzy relations between two sets and two unknown fuzzy relations on these sets. Solutions to these systems are pairs of fuzzy relations on the underlying sets and can be ordered coordinatewise. We show that solutions to each of these systems form a complete lattice, and provide procedures for computing the greatest solution which is less than or equal to a given pair of fuzzy relations. We also demonstrate the application of solutions to these systems in data reduction.
Semilattices of weakly left Archimedean semigroups
Filomat, 1995
By the well-known result of A. H. Clifford, any band of left Archimedean semigroups is a semilatt... more By the well-known result of A. H. Clifford, any band of left Archimedean semigroups is a semilattice of matrices (rectangular bands) of left Archimedean semigroups. The converse of this assertion don’t hold, i.e. the class of semilattices of matrices of left Archimedean semigroups is larger than the class of bands of left Archimedean semigroups. In this paper we characterize semilattices of matrices of left Archimedean semigroups, and especially matrices of left Archimedean semigroups. The obtained results generalize the some results of M. S. Putcha and L. N. Shevrin. Bands of left (also right and two-sided) Archimedean semigroups form important classes of semigroups studied by a number of authors. General characterizations of these semigroups have been given by M. S. Putcha [16], and in the completely π-regular case by L. N. Shevrin [17]. Some new characterizations of bands of left Archimedean semigroups and of bands of nil-extensions of left simple semigroups have been given recently by the authors [6]. By the well-known result of A. H. Clifford, any band of left Archimedean semigroups is a semilattice of matrices (rectangular bands) of left Archimedean semigroups. The converse of this assertion don’t hold, i.e. the class of semilattices of matrices of left Archimedean semigroups is larger than the class of bands of left Archimedean semigroups. In this paper we give a complete characterization of semigroups having a semilattice decomposition whose components are matrices of left Archimedean semigroups. Moreover, we describe such components in the general and some special cases. For the related results see [7], [12] and [13]. For more informations about semilattice-matrix decompositions of semigroups the reader is reffered Supported by Grant 0401B of RFNS through Math. Inst. SANU. 603 604 S. Bogdanović and M. Ćirić to [10] and [11]. The obtained results generalize the above quoted results of M. S. Putcha and L. N. Shevrin. Throughout this paper Z+ will denote the set of positive integers. The division relations | and | l on a semigroup S are defined by a | b ⇔ (∃x, y ∈ S1) b = xay, a | l b ⇔ (∃x ∈ S1) b = xa, and that the relations −→ and l −→ on S are defined by a −→ b ⇔ (∃n ∈ Z+) a | bn, a l −→ b ⇔ (∃n ∈ Z+) a | l bn. The relation r −→ on S is defined dually. For n ∈ Z+, l −→n will denote the n-th power of l −→, and l −→∞ will denote the transitive closure of l −→. For an element a of a semigroup S we define sets Λn(a), n ∈ Z+, and Λ(a) by Λn(a) = {x ∈ S | a l −→n x}, Λ(a) = {x ∈ S | a l −→∞ x}, and the equivalence relations Λn, n ∈ Z+, and λ on S by aλn b ⇔ Λn(a) = Λn(b), a λ b ⇔ Λ(a) = Λ(b), [3]. For undefined notions and notations we refer to [1], [2] and [14]. First we prove the following theorem: Theorem 1. The following conditions on a semigroup S are equivalent: (i) λ is a matrix congruence on S; (ii) λ is a right zero band congruence on S; (iii) (∀a, b, c ∈ S) abc l −→∞ ac; (iv) (∀a, b ∈ S) aba l −→∞ a; (v) (∀a, b ∈ S) ab l −→∞ b; (vi) S is a disjoint union of all its principal left radicals; (vii) l −→∞ is a symmetric relation on S. Proof. (i) ⇒ (iii), (iii) ⇒ (iv) and (ii) ⇒ (i). This follows immediately. (iv) ⇒ (v). For all a, b ∈ S, ab l −→ bab, so by (iv), ab l −→∞ b. (v) ⇒ (ii). Let a, b ∈ S such that a λ b, and x ∈ S. By (v), Λ(ax) = Λ(x) = Λ(bx) and Λ(xa) = Λ(a) = Λ(b) = Λ(xb). Therefore, λ is a congruence. Clearly, it is a right zero band congruence. Semilattices of weakly left Archimedean semigroups 605 (ii) ⇒ (vi). Let S be a right zero band B of semigroups Si, i ∈ B, which are λ-classes of S. Assume a ∈ S. Then a ∈ Si, for some i ∈ B, and since Si is a completely semiprime left ideal of S (Lemma 4 [3]), then Λ(a) ⊆ Si. On the other hand, if b ∈ Si, then b λ a, so b ∈ Λ(b) = Λ(a), whence Si ⊆ Λ(a). Therefore, Λ(a) = Si, so (vi) holds. (vi) ⇒ (vii). Let a, b ∈ S such that a l −→∞ b. Then b ∈ Λ(a), whence Λ(a) ∩ Λ(b) 6 = ∅, so by (vi), Λ(a) = Λ(b). Therefore, b l −→∞ a. (vii) ⇒ (v). For all a, b ∈ S, b l −→ ab, so by (vii), ab l −→∞ b. Corollary 1. The following conditions on a semigroup S are equivalent: (i) λn is a matrix congruence on S; (ii) λn is a right zero band congruence on S; (iii) (∀a, b ∈ S) Λn(a) ⊆ Λn(aba); (iv) (∀a, b ∈ S) Λn(b) ⊆ Λn(ab); (v) l −→n is a symmetric relation on S. Lemma 1. Let ξ be a band congruence on a semigroup S contained in l , where l = l −→ ∩ l −→ −1 . Then any ξ-class of S is a left Archimedean semigroup. Recall that a semigroup S is called left Archimedean if a l −→ b, for all a, b ∈ S. Here we introduce a more general notion: a semigroup S will be called weakly left Archimedean if ab l −→ b, for all a, b ∈ S. Weakly right Archimedean semigroups are defined dually. A semigroup S is weakly tArchimedean (or weakly two-sided Archimedean) if it is both weakly left and weakly right Archimedean, i.e. if for all a, b ∈ S there exists n ∈ Z+ such that an ∈ abSba. We give the following characterization of semilattices of weakly left Archimedean semigroups:…
Proceedings of the International Conference on Algebra 2010 - Advances in Algebraic Structures, 2011
ABSTRACT M. S. Putcha, in 1973, gave the first complete description of semilattices of Archimedea... more ABSTRACT M. S. Putcha, in 1973, gave the first complete description of semilattices of Archimedean semigroups. Other characterizations of this class of semigroups have been given by T. Tamura in 1972, by S. Bogdanović and M. Ćirić in 1992 and by M. Ćirić and S. Bogdanović in 1993. In this paper, for m,n∈N, we define a relation ρ (m,n) and prove that it is a congruence relation on an arbitrary semigroup. Using the congruence relation ρ (m,n) we give some new characterizations of semilattices of Archimedean semigroups. At the end we describe hereditary properties of semilattices of k-Archimedean semigroups.
Proceedings of the Japan Academy, Series A, Mathematical Sciences, 1993
Spined products of semigroups were first defined and studied by N. Kimura, 1958, [7]. After that,... more Spined products of semigroups were first defined and studied by N. Kimura, 1958, [7]. After that, spined products have been considered many a time, predominantly those of a band and a semilattice of semigroups with respect to their common semilattice homomorphic image. Spined and subdirect products of a band and a semilattice of groups are studied by M. Yamada [13], [14], J. M. Howie and G. Lallement [6] and by M. Petrich [10]; spined products of a band and some types of semilattices of monoids are studied by F. Pastijn [8], A. E1-Qallali [3], [4], and by R. J. Warne [12]. For other considerations of these products, we refer to [4], [5], [7], [9], [15]. In the quoted papers, spined products are considered in connection with some types of bands of semigroups. In this paper, we give a general composition for bands of semigroups that are (punched) spined products of a band and a semilattice of semigroups. This composition, in some sense, is a generalization of a well-known semilattice composition (see Theorem III 7.2. [9]). Let B be a band. By-<1 and N we denote quasi-orders on B defined by i < je* ij j, i <,.j <:> ji j, and by _< we denote the natural order on
Recently, two types of simulations (forward and backward simulations) and four types of bisimulat... more Recently, two types of simulations (forward and backward simulations) and four types of bisimulations (forward, backward, forward-backward, and backward-forward bisimulations) between fuzzy automata have been introduced. If there is at least one simulation/bisimulation of some of these types between the given fuzzy automata, it has been proved that there is the greatest simulation/bisimulation of this kind. In the present paper, for any of the above-mentioned types of simulations/bisimulations we provide an effective algorithm for deciding whether there is a simulation/bisimulation of this type between the given fuzzy automata, and for computing the greatest one, whenever it exists. The algorithms are based on the method developed in [J. Ignjatović, M.Ćirić, S. Bogdanović, On the greatest solutions to certain systems of fuzzy relation inequalities and equations, Fuzzy Sets and Systems 161 (2010) 3081-3113], which comes down to the computing of the greatest post-fixed point, contained in a given fuzzy relation, of an isotone function on the lattice of fuzzy relations.
The Existence of Generalized Inverses of Fuzzy Matrices
Studies in Computational Intelligence, 2018
In this paper we show that every fuzzy matrix with entries in a complete residuated lattice posse... more In this paper we show that every fuzzy matrix with entries in a complete residuated lattice possess the generalized inverses of certain types, and in particular, it possess the greatest generalized inverses of these types. We also provide an iterative method for computing these greatest generalized inverses, which terminates in a finite number of steps, for example, for all fuzzy matrices with entries in a Heyting algebra. For other types of generalized inverses we determine criteria for the existence, given in terms of solvability of particular systems of linear matrix equations. When these criteria are met, we prove that there is the greatest generalized inverse of the given type and provide a direct method for its computing.
In this paper we provide procedures for testing the existence of various types of generalized inv... more In this paper we provide procedures for testing the existence of various types of generalized inverses in involutive residuated semigroups and involutive quantales defined by the Moore-Penrose equations, and for computing the extreme inverses whenever they exist. We also determine certain instances when a? = a*, whenever a? exists. The obtained results can be applied to a wide class of quantales of fuzzy relations and fuzzy matrices, as well as to Gelfand quantales.
In this paper we provide improvements of determinization methods, based on factorization of fuzzy... more In this paper we provide improvements of determinization methods, based on factorization of fuzzy states, for fuzzy finite automata that accept fuzzy languages of infinite range. The improvements are based on the usage of the fuzzy relational calculus, namely, on the usage of the right invariant fuzzy quasi-orders. Our algorithms perform better in the sense that they produce smaller automata, while require the same computation time. In addition, they can produce finite deterministic automata in cases when previous algorithms result in infinite deterministic automata. We show that the weak representable-cycles property is necessary and sufficient condition for determinization of a fuzzy automaton via a maximal factorization of fuzzy states. This condition is more general than the representable-cycles property previously determined as the necessary and sufficient condition for determinization of a fuzzy automaton via a maximal factorization of fuzzy states.
In this paper we consider different types of ranks of fuzzy matrices over residuated lattices. We... more In this paper we consider different types of ranks of fuzzy matrices over residuated lattices. We investigate relations between ranks and prove that row rank, column rank and Schein rank of idempotent fuzzy matrices are equal. In particular, ranks and corresponding decompositions of fuzzy matrices representing fuzzy quasi-orders are studied in detail. We show that fuzzy matrix decomposition by ranks can be used in the state reduction of fuzzy automata. Moreover, we prove that using rank decomposition of fuzzy matrices improves results of any state reduction method based on merging indistinguishable states of fuzzy automata.
In this paper we study several types of systems of fuzzy relation equations and inequalities comp... more In this paper we study several types of systems of fuzzy relation equations and inequalities composed of a given family of fuzzy relations between two sets and two unknown fuzzy relations on these sets. Solutions to these systems are pairs of fuzzy relations on the underlying sets and can be ordered coordinatewise. We show that solutions to each of these systems form a complete lattice, and provide procedures for computing the greatest solution which is less than or equal to a given pair of fuzzy relations. We also demonstrate the application of solutions to these systems in data reduction.
Semilattices of weakly left Archimedean semigroups
Filomat, 1995
By the well-known result of A. H. Clifford, any band of left Archimedean semigroups is a semilatt... more By the well-known result of A. H. Clifford, any band of left Archimedean semigroups is a semilattice of matrices (rectangular bands) of left Archimedean semigroups. The converse of this assertion don’t hold, i.e. the class of semilattices of matrices of left Archimedean semigroups is larger than the class of bands of left Archimedean semigroups. In this paper we characterize semilattices of matrices of left Archimedean semigroups, and especially matrices of left Archimedean semigroups. The obtained results generalize the some results of M. S. Putcha and L. N. Shevrin. Bands of left (also right and two-sided) Archimedean semigroups form important classes of semigroups studied by a number of authors. General characterizations of these semigroups have been given by M. S. Putcha [16], and in the completely π-regular case by L. N. Shevrin [17]. Some new characterizations of bands of left Archimedean semigroups and of bands of nil-extensions of left simple semigroups have been given recently by the authors [6]. By the well-known result of A. H. Clifford, any band of left Archimedean semigroups is a semilattice of matrices (rectangular bands) of left Archimedean semigroups. The converse of this assertion don’t hold, i.e. the class of semilattices of matrices of left Archimedean semigroups is larger than the class of bands of left Archimedean semigroups. In this paper we give a complete characterization of semigroups having a semilattice decomposition whose components are matrices of left Archimedean semigroups. Moreover, we describe such components in the general and some special cases. For the related results see [7], [12] and [13]. For more informations about semilattice-matrix decompositions of semigroups the reader is reffered Supported by Grant 0401B of RFNS through Math. Inst. SANU. 603 604 S. Bogdanović and M. Ćirić to [10] and [11]. The obtained results generalize the above quoted results of M. S. Putcha and L. N. Shevrin. Throughout this paper Z+ will denote the set of positive integers. The division relations | and | l on a semigroup S are defined by a | b ⇔ (∃x, y ∈ S1) b = xay, a | l b ⇔ (∃x ∈ S1) b = xa, and that the relations −→ and l −→ on S are defined by a −→ b ⇔ (∃n ∈ Z+) a | bn, a l −→ b ⇔ (∃n ∈ Z+) a | l bn. The relation r −→ on S is defined dually. For n ∈ Z+, l −→n will denote the n-th power of l −→, and l −→∞ will denote the transitive closure of l −→. For an element a of a semigroup S we define sets Λn(a), n ∈ Z+, and Λ(a) by Λn(a) = {x ∈ S | a l −→n x}, Λ(a) = {x ∈ S | a l −→∞ x}, and the equivalence relations Λn, n ∈ Z+, and λ on S by aλn b ⇔ Λn(a) = Λn(b), a λ b ⇔ Λ(a) = Λ(b), [3]. For undefined notions and notations we refer to [1], [2] and [14]. First we prove the following theorem: Theorem 1. The following conditions on a semigroup S are equivalent: (i) λ is a matrix congruence on S; (ii) λ is a right zero band congruence on S; (iii) (∀a, b, c ∈ S) abc l −→∞ ac; (iv) (∀a, b ∈ S) aba l −→∞ a; (v) (∀a, b ∈ S) ab l −→∞ b; (vi) S is a disjoint union of all its principal left radicals; (vii) l −→∞ is a symmetric relation on S. Proof. (i) ⇒ (iii), (iii) ⇒ (iv) and (ii) ⇒ (i). This follows immediately. (iv) ⇒ (v). For all a, b ∈ S, ab l −→ bab, so by (iv), ab l −→∞ b. (v) ⇒ (ii). Let a, b ∈ S such that a λ b, and x ∈ S. By (v), Λ(ax) = Λ(x) = Λ(bx) and Λ(xa) = Λ(a) = Λ(b) = Λ(xb). Therefore, λ is a congruence. Clearly, it is a right zero band congruence. Semilattices of weakly left Archimedean semigroups 605 (ii) ⇒ (vi). Let S be a right zero band B of semigroups Si, i ∈ B, which are λ-classes of S. Assume a ∈ S. Then a ∈ Si, for some i ∈ B, and since Si is a completely semiprime left ideal of S (Lemma 4 [3]), then Λ(a) ⊆ Si. On the other hand, if b ∈ Si, then b λ a, so b ∈ Λ(b) = Λ(a), whence Si ⊆ Λ(a). Therefore, Λ(a) = Si, so (vi) holds. (vi) ⇒ (vii). Let a, b ∈ S such that a l −→∞ b. Then b ∈ Λ(a), whence Λ(a) ∩ Λ(b) 6 = ∅, so by (vi), Λ(a) = Λ(b). Therefore, b l −→∞ a. (vii) ⇒ (v). For all a, b ∈ S, b l −→ ab, so by (vii), ab l −→∞ b. Corollary 1. The following conditions on a semigroup S are equivalent: (i) λn is a matrix congruence on S; (ii) λn is a right zero band congruence on S; (iii) (∀a, b ∈ S) Λn(a) ⊆ Λn(aba); (iv) (∀a, b ∈ S) Λn(b) ⊆ Λn(ab); (v) l −→n is a symmetric relation on S. Lemma 1. Let ξ be a band congruence on a semigroup S contained in l , where l = l −→ ∩ l −→ −1 . Then any ξ-class of S is a left Archimedean semigroup. Recall that a semigroup S is called left Archimedean if a l −→ b, for all a, b ∈ S. Here we introduce a more general notion: a semigroup S will be called weakly left Archimedean if ab l −→ b, for all a, b ∈ S. Weakly right Archimedean semigroups are defined dually. A semigroup S is weakly tArchimedean (or weakly two-sided Archimedean) if it is both weakly left and weakly right Archimedean, i.e. if for all a, b ∈ S there exists n ∈ Z+ such that an ∈ abSba. We give the following characterization of semilattices of weakly left Archimedean semigroups:…
Proceedings of the International Conference on Algebra 2010 - Advances in Algebraic Structures, 2011
ABSTRACT M. S. Putcha, in 1973, gave the first complete description of semilattices of Archimedea... more ABSTRACT M. S. Putcha, in 1973, gave the first complete description of semilattices of Archimedean semigroups. Other characterizations of this class of semigroups have been given by T. Tamura in 1972, by S. Bogdanović and M. Ćirić in 1992 and by M. Ćirić and S. Bogdanović in 1993. In this paper, for m,n∈N, we define a relation ρ (m,n) and prove that it is a congruence relation on an arbitrary semigroup. Using the congruence relation ρ (m,n) we give some new characterizations of semilattices of Archimedean semigroups. At the end we describe hereditary properties of semilattices of k-Archimedean semigroups.
Proceedings of the Japan Academy, Series A, Mathematical Sciences, 1993
Spined products of semigroups were first defined and studied by N. Kimura, 1958, [7]. After that,... more Spined products of semigroups were first defined and studied by N. Kimura, 1958, [7]. After that, spined products have been considered many a time, predominantly those of a band and a semilattice of semigroups with respect to their common semilattice homomorphic image. Spined and subdirect products of a band and a semilattice of groups are studied by M. Yamada [13], [14], J. M. Howie and G. Lallement [6] and by M. Petrich [10]; spined products of a band and some types of semilattices of monoids are studied by F. Pastijn [8], A. E1-Qallali [3], [4], and by R. J. Warne [12]. For other considerations of these products, we refer to [4], [5], [7], [9], [15]. In the quoted papers, spined products are considered in connection with some types of bands of semigroups. In this paper, we give a general composition for bands of semigroups that are (punched) spined products of a band and a semilattice of semigroups. This composition, in some sense, is a generalization of a well-known semilattice composition (see Theorem III 7.2. [9]). Let B be a band. By-<1 and N we denote quasi-orders on B defined by i < je* ij j, i <,.j <:> ji j, and by _< we denote the natural order on
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