Search Results (12)

An involutive m-semilattice is a kind of algebraic structure with symmetry. Symmetry is reflected from partial-order relations to algebraic operations and even categorical properties. In this study, firstly, the concepts of the nucleus and congruence in involutive m-semilattices are introduced, and their interrelationships are discussed. On this basis, the concrete structure of a coequalizer in the category of involutive m-semilattices is obtained. We introduce the definition of free involutive m-semilattices, and the concrete structure of involutive m-semilattices is discussed. In addition, It is shown that the category of involutive m-semilattices is algebraic. Secondly, the colimit in the category of involutive m-semilattices is shown to be a very difficult problem. We obtain the concrete structure of the colimit for a full subcategory of the category of involutive m-semilattices. Thirdly, we introduce the definition of an inverse system in the category of involutive m-semilattices and give the concrete structure of the inverse limit of an inverse system. We establish the concept of a mapping between two inverse systems. The properties of inverse limits are discussed. Finally, we study the direct limit of the category of involutive m-semilattices and give its concrete structure. Full article

Basketball games are characterized by the large number of lineups that can be used by the coach during a game, e.g., with 12 players there are 792 possible lineups. This has led to the development of several statistics for the combinations of players on the court since team performance depends on synergy among players. It is of primary importance for a basketball team to understand the team performance and aid the coaching staff in making the proper decisions. In this work, we apply data mining and knowledge extraction techniques to basketball analytics. In particular, we propose an algorithm for answering questions (including filtering and maximization) about the team performance when any K-Player lineup is on the court ($1\le K\le 5$). The algorithm uses a semilattice representation and a depth-first search traversal that incrementally computes the statistics by exploiting set theory properties. As a case study, we provide experiments by using lineups mainly from the EuroLeague Basketball but also from the National Basketball Association (NBA). Regarding the results, the proposed method is more than 30× faster than the baseline for the EuroLeague and 200× faster for the NBA. Indicatively, we can compute the key traditional cumulative and average statistics for all K-Player combinations of players of the EuroLeague of a single season in less than 1 s. Finally, we introduce indicative statistics using the computations mentioned. Full article

A semigroup S is termed a generalized Clifford semigroup (GC-semigroup) if it forms a strong semilattice of $\pi$-groups. This paper explores necessary and sufficient conditions for a GC-monoid to be nearly complete within certain subclasses. These subclasses are distinguished by the nature of their linking homomorphisms, which may be bijective, surjective, injective, or image trivial. The findings provide a deeper understanding of the structural integrity and completeness of GC-monoids, contributing valuable insights to the theoretical framework of semigroup theory. Applications of this study span various fields, including cryptography for secure algorithm design, coding theory and quantum computing for advanced quantum algorithms. The established criteria also support further research in mathematical biology and automorphic theory, demonstrating the broad relevance and utility of nearly complete GC-monoids. Full article

Balancing the accuracy and the complexity of models is a well established and ongoing challenge. Models can be misleading if they are not accurate, but models may be incomprehensible if their accuracy depends upon their being complex. In this paper, semilattices are examined as an option for balancing the accuracy and the complexity of machine learning models. This is done with a type of machine learning that is based on semilattices: algebraic machine learning. Unlike trees, semilattices can include connections between elements that are in different hierarchies. Trees are a subclass of semilattices. Hence, semilattices have higher expressive potential than trees. The explanation provided here encompasses diagrammatic semilattices, algebraic semilattices, and interrelationships between them. Machine learning based on semilattices is explained with the practical example of urban food access landscapes, comprising food deserts, food oases, and food swamps. This explanation describes how to formulate an algebraic machine learning model. Overall, it is argued that semilattices are better for balancing the accuracy and complexity of models than trees, and it is explained how algebraic semilattices can be the basis for machine learning models. Full article

This paper introduces a representation of subnetworks of a network $\Gamma$ consisting of a set of vertices and a set of relations, where relations are the primitive structures of a network. It is proven that all connected subnetworks of a network $\Gamma$ form a quasi-semilattice $\mathcal{L}(\Gamma )$, namely a network quasi-semilattice.Two equivalences $\sigma$ and $\delta$ are defined on $\mathcal{L}(\Gamma )$. Each $\delta$ class forms a semilattice and also has an order structure with the maximum element and minimum elements. Here, the minimum elements correspond to spanning trees in graph theory. Finally, we show how graph inverse semigroups, Leavitt path algebras and Cuntz–Krieger graph $C^{*}$-algebras are constructed in terms of relations. Full article

The “semilattices of Mal’cev blocks”, for short SMB algebras, were defined by A. Bulatov. In a recently accepted paper by P. Đapić, P. Marković, R. McKenzie, and A. Prokić, the class of all SMB algebras and its subclass of regular SMB algebras were proved to be varieties of algebras. In this paper, we find an equational base of the first variety and simplify the previously known equational base of the second variety. Full article

A subset A of a semigroup S is called a chain (antichain) if $ab\in \{a,b\}$ ($ab\notin \{a,b\}$) for any (distinct) elements $a,b\in A$. A semigroup S is called periodic if for every element $x\in S$ there exists $n\in \mathbb{N}$ such that $x^n$ is an idempotent. A semigroup S is called (anti)chain-finite if S contains no infinite (anti)chains. We prove that each antichain-finite semigroup S is periodic and for every idempotent e of S the set $\sqrt[\infty ]{e}=\{x\in S:\exists n\in \mathbb{N}(x^n=e)\}$ is finite. This property of antichain-finite semigroups is used to prove that a semigroup is finite if and only if it is chain-finite and antichain-finite. Furthermore, we present an example of an antichain-finite semilattice that is not a union of finitely many chains. Full article

In business, managers may use the association information among products to define promotion and competitive strategies. The mining of high-utility association rules (HARs) from high-utility itemsets enables users to select their own weights for rules, based either on the utility or confidence values. This approach also provides more information, which can help managers to make better decisions. Some efficient methods for mining HARs have been developed in recent years. However, in some decision-support systems, users only need to mine a smallest set of HARs for efficient use. Therefore, this paper proposes a method for the efficient mining of non-redundant high-utility association rules (NR-HARs). We first build a semi-lattice of mined high-utility itemsets, and then identify closed and generator itemsets within this. Following this, an efficient algorithm is developed for generating rules from the built lattice. This new approach was verified on different types of datasets to demonstrate that it has a faster runtime and does not require more memory than existing methods. The proposed algorithm can be integrated with a variety of applications and would combine well with external systems, such as the Internet of Things (IoT) and distributed computer systems. Many companies have been applying IoT and such computing systems into their business activities, monitoring data or decision-making. The data can be sent into the system continuously through the IoT or any other information system. Selecting an appropriate and fast approach helps management to visualize customer needs as well as make more timely decisions on business strategy. Full article

Conditions that are necessary for the relative annihilator in lower $BCK$ -semilattices to be a prime ideal are discussed. Given the minimal prime decomposition of an ideal A, a condition for any prime ideal to be one of the minimal prime factors of A is provided. Homomorphic image and pre-image of the minimal prime decomposition of an ideal are considered. Using a semi-prime closure operation “ $cl$ ”, we show that every minimal prime factor of a $cl$ -closed ideal A is also $cl$ -closed. Full article

In any logical algebraic structures, by using of different kinds of filters, one can construct various kinds of other logical algebraic structures. With this inspirations, in this paper by considering a hoop algebra or a hoop, that is introduced by Bosbach, the notion of co-filter on hoops is introduced and related properties are investigated. Then by using of co-filter, a congruence relation on hoops is defined, and the associated quotient structure is studied. Thus Brouwerian semilattices, Heyting algebras, Wajsberg hoops, Hilbert algebras and BL-algebras are obtained. Full article

In this paper, we introduce the notions of $(\in ,\in )$ -fuzzy positive implicative filters and $(\in ,\in \vee q)$ -fuzzy positive implicative filters in hoops and investigate their properties. We also define some equivalent definitions of them, and then we use the congruence relation on hoop defined in blue[Aaly Kologani, M.; Mohseni Takallo, M.; Kim, H.S. Fuzzy filters of hoops based on fuzzy points. Mathematics. 2019, 7, 430; doi:10.3390/math7050430] by using an $(\in ,\in )$ -fuzzy filter in hoop. We show that the quotient structure of this relation is a Brouwerian semilattice. Full article

We study entropy inequalities for variables that are related by functional dependencies. Although the powerset on four variables is the smallest Boolean lattice with non-Shannon inequalities, there exist lattices with many more variables where the Shannon inequalities are sufficient. We search for conditions that exclude the existence of non-Shannon inequalities. The existence of non-Shannon inequalities is related to the question of whether a lattice is isomorphic to a lattice of subgroups of a group. In order to formulate and prove the results, one has to bridge lattice theory, group theory, the theory of functional dependences and the theory of conditional independence. It is demonstrated that the Shannon inequalities are sufficient for planar modular lattices. The proof applies a gluing technique that uses that if the Shannon inequalities are sufficient for the pieces, then they are also sufficient for the whole lattice. It is conjectured that the Shannon inequalities are sufficient if and only if the lattice does not contain a special lattice as a sub-semilattice. Full article