In this article, after the definitions of the reduced hesitant L-fuzzy automaton (RHLFA) and the minimal hesitant L-fuzzy automaton; we convert a hesitant L-fuzzy automaton (HLFA) to a RHLFA by reducing the number of its states such that its language is equal to the original HLFA language. Then, by defining an equivalence relation on the monoid X & lowast; , we construct an HLFA whose language is equal to the language of the transformed RHLFA, and we show that this HLFA is minimal. In conclusion, we delineate the criteria under which, the number of states in the minimal HLFA is equal to the number of states in the RHLFA.

In this study, we show that automata theory is also a suitable tool for analyzing a more complex type of the k-forcing process. First, the definition of k-forcing automata is presented according to the definition of k-forcing for graphs. Moreover, we study and discuss the language of k-forcing automata for particular graphs. Also, for some graphs with different k-forcing sets, we study the languages of their k-forcing automata. In addition, for some given recognizable languages, we study the structure of graphs. After that, we show that k-forcing automata arising from isomorph graphs are also isomorph. Also, we present the style of words that can be recognized with k-forcing automata. Moreover, we introduce the structure of graphs the k-forcing automata arising from which recognize some particular languages. To clarify the notions and the results obtained in this study, some examples are submitted as well.

In this paper, we study a new generalization for the notion of fuzzy automata, which we called hesitant L-fuzzy automaton (HLFA). We present the formulations of the mathematics framework for the theory of HLFA. Moreover, we present the concepts of hesitant L-fuzzy behavior and inverse hesitant L-fuzzy behavior recognized by a type of HLFA. After that, for any hesitant L-fuzzy language we present a minimal complete accessible deterministic hesitant L-fuzzy automaton recognizing that. Finally, we present an algorithm, which determines states of the minimal hesitant L-fuzzy automaton and we present the time complexity of the algorithm.

This study aims to investigate the characterization of algebraic concepts of subsystem, retrievability and connectivity of an L-valued tree transition system based on its related layers. It also seeks to associate upper semilattices with L-valued tree transition systems. Further, a decomposition of an L-valued tree transition system is provided in terms of its layers. In addition, it proposes a construction of an Lvalued tree transition system which corresponds to a given finite poset. Then an isomorphism is established between the poset of class of subsystem of an L-valued tree transition system and an upper semilattice.

This study aims to investigate the algebraic properties of fuzzy multiset finite automaton (FMFA), giving a congruence relation. Using the equivalence classes, a minimal accessible complete FMFA is then presented. In addition, we define the concepts of admissible relation, admissible partition for FMFA and quotient FMFA. Further, we present an algorithm which determines an admissible partition for an FMFA and we suggest the time complexity of the algorithm. In particular, we introduce a connection between the admissible partition and the quotient FMFA and we show that any quotient of a given FMFA and the FMFA itself have the same language. Furthermore, using the quotient FMFA, we obtain an irreducible FMFA with the same language.

Semi-supervised clustering, utilizing the supervision information to guide the clustering process, could improve the clustering effect of the models. Most of existing semi-supervised clustering models only consider pairwise constraints or pointwise constraints. In this paper, the semi-supervised method is applied to the fuzzy clustering algorithm, and a robust semi-supervised fuzzy clustering algorithm is proposed. Firstly, fully considering prior knowledge, our models integrate pointwise constraints and pairwise constraints into a unified framework to improve the clustering performance of the fuzzy clustering algorithm. Secondly, in order to alleviate the impact of outliers, the robust performance of the models is considered by introducing an adaptive loss function into the models. Thirdly, our models can capture the global structures and the local manifold structures of data sets. Finally, a simple and efficient algorithm is proposed to solve the models, which ensures that the obtained solution is sparse and satisfies the constraint conditions in our models. Compared with five representative methods, experimental results on public datasets, such as text dataset (dbworld), show the effectiveness of the proposed models.

In this paper, we show that for any BL-general L-fuzzy au-tomaton (BL-GLFA) there exists a complete deterministic accessible re-duced BL-general L-fuzzy automaton that recognizing the behavior of the BL-GLFA. Also, we prove that for any finite realization ,9, there exists a minimal complete deterministic BL-GLFA recognizing ,9. We prove any complete deterministic accessible reduced BL-GLFA is a minimal BL-GLFA. After that, we show that for any given finite realization ,9, the minimal complete deterministic BL-GLFA recognizing ,9 is isomorphic to any complete accessible deterministic reduced BL-GLFA recognizing ,9. Moreover, we give some examples to clarify these notions. Finally, by using these notions, we give some theorems and algorithms and obtain some related results.

In recent years, the rapid development of quantum computation has stimulated researchers to establish a theory of computation based on quantum logic. The present study therefore aims to introduce the notion of lattice-valued general orthomodular automaton, for simplicity, lattice-valued GOA. The class of languages, accepted by lattice-valued GOA, is also defined and explicated. Moreover, the acceptance abilities of lattice-valued GOA and their various modifications are compared. Finally, the closure properties of orthomodular languages as union, complement and product are consequently derived.

A new strongly regular relation theta*n is defined on polygroup P such that the quotient P/theta*n, the set of all equivalence classes, is a Bell group for n E {2, 3}.

This paper defines a novel graph constructed on a residuated lattice using a specific definition of the fuzzy graph (called L-graph). These types of graphs also have applications in libraries, pharmacies and machine facilities. Among these applications, this research presents the L-graph automaton related to the L-graph constructed by a minimum zero forcing set. The L-graph automaton is used to find efficient searching for diseases that have the most similar symptoms. It has been demonstrated that if two L-graphs are isomorphic, then the related L-graph automata are isomorphic but not necessarily vice versa. New notions, L ''-graph automata, pseudo-isomorphism and self-sufficiency are proved. Besides, by taking advantage of the properties of the residuated lattice, it has been proven that the two L-graph automata are equivalent if and only if the L' -graph automata contracted of L -graph automata are equivalent. Ultimately, in light of the above, some related theorems are proved and several examples are provided to illustrate these new notions.

The current study aims to investigate the L-valued tree automata theory based on t-norm/t-conorm and it further examines their algebraic and L-valued topological properties. Specifically, the concept of L-valued operators with t-norm/t-conorm is introduced and the existing relationships between them are also studied. Interestingly, we associate Lvalued co-topologies/topologies for a given L-valued tree automaton, using them to characterize some algebraic concepts. Further, we introduce the concepts of Alexandroff L-graded co-topologies and Alexandroff L-graded topologies which correspond to the L-valued operators with t-norm and L-valued operators with t-conorm/implicator, respectively. In addition, we aim to specify the relationship between the L-graded co-topologies/topologies, showing that the introduced L-graded co-topologies/topologies have some interesting consequences under homomorphism.

‎This paper introduces the notion of the combined product of two‎ ‎$RL$-graphs‎ ‎while it is an‎ ‎$RL$-graph‎. ‎It is stated that the combined product has commutative properties‎, ‎i.e.‎, ‎$G\boxtimes H$‎ ‎and‎ ‎$H\boxtimes G$ are two isomorphic $RL$-graphs‎. ‎Moreover‎, ‎it is shown under a theorem that two isomorphic‎ ‎$RL$-graphs‎ ‎$G$‎ ‎and‎ ‎$G'$‎ ‎and‎ ‎two isomorphic‎ ‎$RL$-graphs‎ ‎$H$‎ ‎and‎ ‎$H'$‎ ‎have isomorphic combined products‎ ‎$G\boxtimes H$‎ ‎and‎ ‎$G'\boxtimes H'$‎. ‎Further‎, ‎it is investigated the relationships between these graphs and their operations by some notions such as strong‎ ‎$RL$-graph‎, ‎regular $RL$-graph‎, ‎and totally regular $RL$-graph‎. ‎Afterward‎, ‎it is displayed in a theorem that the combined product of two regular ($\alpha$-regular‎, ‎complete‎, ‎connected) $RL$-graphs is a regular ($\alpha$-regular‎, ‎complete‎, ‎connected) $RL$-graph‎. ‎It is also shown in theorems what properties certain types of‎ ‎$RL$-graphs combined will have‎. ‎Also‎, ‎these notions and theorems are clarified by some examples‎. ‎The combined product of two‎ ‎$RL$-graphs has many applications in various fields‎, ‎such as probability sciences‎, ‎urban planning‎, ‎etc‎. ‎In this article‎, ‎only two of these applications‎, ‎which determine the impact of effective factors on people's quality of life‎ ‎and‎ ‎factors effective in raising the production of a factory‎, ‎are stated and they are clarified by an example‎.

In this paper, we define the concepts of single-valued neutrosophic general automaton, complete and deterministic single-valued neutrosophic general automaton. We present a minimal single-valued neutrosophic general automaton that preserves the language for a given single-valued neutrosophic general automaton. Moreover, we present the closure properties such as union and intersection for single-valued neutrosophic general automata.

This study aimed to examine deterministic FMFA, complete FMFA, and accessible FMFA, using the definition of fuzzy multiset finite automaton (FMFA). We show that a fuzzy multiset language is accepted by a fuzzy multiset finite automaton if and only if it is accepted by an accessible deterministic complete fuzzy multiset finite automaton. We also introduce and study the minimal FMFA (minimal accessible deterministic complete fuzzy multiset finite automata) for a given fuzzy multiset language. In addition, we present a reduced accessible deterministic complete FMFA for a given multiset regular language. Finally, we show that both the minimal FMFA and the reduced accessible deterministic complete FMFA are isomorphic.

The current study aimed to describe the notion of the L-graph, which is constructed on a residuated lattice, presenting the idea of the maximal product of two L-graphs. Moreover, the applied algorithm demonstrated its efficiency to a great extent to improve the educational system. The results showed that the maximal product of both G and H and reversely, H and G are isomorphic L-graphs. The maximal product of two L-graph automata was represented as a new notion, followed by investigating their self-sufficiency conditions in two different modes. In addition, A(Z(G)) * A(Z(H)) and A(Z(H)) * A(Z(G)) were proved as two isomorphic-related L-graph automata considering the residuated lattice properties. Then, correlations among A(Z(G)) * A(Z(H)), A(Z(G)), and A(Z(H)) behaviors were explained. As a result, some theorems of the relationship between L-graph automata and their maximal product were introduced. Subsequently, some related theorems were proved, and several examples were provided to illustrate these new notions. Further, an application of the maximal product of two related L-graph automata was expressed in the factors affecting the spared of the coronavirus, and accordingly, some solutions were suggested to reduce the spread of the virus.

This paper first introduces the L-graph automaton built on an L-graph using a zero-constraint set. Then, we study the behavior of the corresponding automaton when the L-graph is a path L-graph (cycle, complete, fully bipartite). These L-graph automata have some applications in various fields. One of them is the identification of drugs that have the most similar side effects. These concepts and applications have been illustrated with some examples.

This paper introduces a graph built on a residual lattice called the L graph. It determines the notion of the strong L-graph. It introduces new concepts, such as direct sum and direct product of two L-graphs. The L-graphs have many applications. For instance, they help determine the minimum number of hospitals equipped with all the treatment wards needed. Accordingly, some related theorems have been proven and several examples have been provided to illustrate these new concepts.

Using the kronecker product definition of two simple graphs, the kronecker product of two RL-graphs was defined and is defined and it is further shown to be an RL-graph. Ultimately, in light of the above, some related theorems are proved and several examples are provided to illustrate these new notions.

This study aims to develop the notion of general fuzzy automata (GFA) to a new one which is known as "LB-valued general fuzzy automata". Instead of the term LB-valued general fuzzy automata, for simplicity, LB-valued GFA is used where B is regarded as a set of propositions about the GFA, in which its underlying structure has been a complete infinitely distributive lattice. Further, LB-valued GFA is scrutinized via different operators and also the interrelationship among these operators is examined. Specifically, it is shown that LB-valued successor, LB-valued predecessor and LB-valued residuated operators play an important role in the algebraic study of LB-valued general fuzzy automaton and that under certain conditions these operators are interrelated. In addition, the concept of a homomorphism between LB-valued general fuzzy automata is introduced and studied. Finally, the concepts of equivalence and congruence are defined, the quotient LB-valued GFA with respect to congruence is formulated and the equivalence between LB-valued GFA and its quotient automaton is proved. To clarify the notions and the results obtained in this study, some examples are submitted as well.

The notions of double-framed soft hyper BCK-algebra and double-framed soft (strong) hyper BCK-ideal of a hyper BCK-algebra are introduced, and related properties are investigated. Characterizations of double-framed soft hyper BCK-algebra and double-framed soft hyper BCK-ideal are considered. Relations between double-framed soft hyper BCK-ideal and double-framed soft strong hyper BCK-ideal are discussed. Conditions for a double-framed soft set to be a double-framed soft strong hyper BCK-ideal are provided.