Maximal Clones Research Articles

Longitudinal sequencing of T cell receptor revealed radiotherapeutic efficacy and prognosis in non-small cell lung cancer patients with brain metastasis.

e21056 Background: Liquid biopsy has played a unique role in long-term surveillance of brain metastasis in non-small cell lung cancer (NSCLC), due to the inaccessibility of solid metastatic biopsies. T cell-mediated immunity is critical in the oncogenesis and metastasis of NSCLC, and sequencing in T cell receptor (TCR) provides a robust method to quantify T cell diversity. Therefore, we aimed to explore the alteration of TCR through liquid biopsy before and after radiotherapy of NSCLC brain metastases. Methods: Thirty NSCLC patients with brain metastases receiving brain radiotherapy were enrolled in this study. Cerebrospinal fluid (CSF) and peripheral blood were collected at baseline, 24 hours (T0) and 28 days (T28) after treatment. TCR sequences were identified by high-throughput sequencing in both compartments. Results: At baseline, dimension reduction analysis identified distinct signatures of V and J gene recombination in blood and CSF TCR sequences. Throughout treatment, both compartments experienced a TCR diversity decrease, however, the degradation of low-abundance clones and the expansion of emerging clones might be two separate processes underwent in blood and CSF, respectively. Diversity changes in blood were possibly related to pulmonary responses, while the increase of maximal clone abundance in CSF might indicate a favorable intracranial response. Superior blood TCR diversity at T28 against baseline was associated with longer OS (HR = 5.700, p = 0.039). Patients with increase of maximal clone abundance ≥ 50 in CSF also had a better intracranial PFS (HR = 8.320, p = 0.011). The predictive effects of these markers were independent of other clinical factors in multivariate Cox analysis. Conclusions: CSF and peripheral blood were independent compartments showing disparate TCR signatures. Longitudinal surveillance of both compartments could be a promising method to predict clinical outcomes for NSCLC patients with brain metastases.

Longitudinal sequencing of TCR and circulating tumor DNA revealing radiotherapeutic efficacy and prognosis for non-small cell lung cancer patients with brain metastasis.

e21181 Background: This study aimed to explore cerebrospinal fluid (CSF) and peripheral blood-based liquid biopsy before and after radiotherapy of non-small cell lung cancer (NSCLC) brain metastases. Methods: Thirty NSCLC patients with brain metastases receiving brain radiotherapy were enrolled in this study. CSF and peripheral blood were collected at baseline, 24 hours (T0) and 28 days (T28) after treatment. Somatic mutations and T-cell receptor (TCR) sequences were identified by high-throughput sequencing in both compartments. Results: At baseline, identical mutations shared by paired blood and CSF emerged in nine patients (30%). Dimension reduction analysis identified distinct signatures of V and J gene recombination in blood and CSF TCR sequences. Throughout treatment, both compartments experienced a TCR diversity decrease, however, the degradation of low-abundance clones and the expansion of emerging clones might be two separate processes underwent in blood and CSF, respectively. Diversity changes and ctDNA in blood were possibly related to pulmonary responses, while the increase of maximal clone abundance in CSF might indicate a favorable intracranial response. Blood-ctDNA clearance at T28 was significantly associated with better overall survival (OS, HR = 4.027, p = 0.028) and progression-free survival (PFS, HR = 4.176, p = 0.024), and superior blood TCR diversity at T28 against baseline was associated with longer OS (HR = 5.700, p = 0.039). Combined blood factors achieved a better predictive power (OS, HR = 10.53, p < 0.001; PFS, HR = 4.843, p < 0.001). Patients with increase of maximal clone abundance ≥ 50 in CSF also had a better intracranial PFS (HR = 8.320, p = 0.011). The predictive effects of these markers were independent of other clinical factors in multivariate Cox analysis. Conclusions: CSF and peripheral blood were independent compartments showing disparate genomic and immune signatures. Longitudinal surveillance of both compartments could be a promising method to predict clinical outcomes for NSCLC patients with brain metastases.

On closed classes in partial k-valued logic that contain the class of monotone functions

Abstract Let A be a precomplete class (a maximal clone) in k-valued logic and T(A) be the family of all closed classes (under superposition) in partial k-valued logic that contain A. A simple test is put forward capable of finding out from a partial order defining the precomplete class A of monotone functions whether the family T(A) is finite or infinite. This completes the solution of the problem of finiteness of T(A) for all precomplete classes of k-valued logic. The proof depends on new families of closed classes founded by the author of the present paper.

Meet-reducible submaximal clones determined by two central relations

Let [Formula: see text] and [Formula: see text] be two central relations on a finite set [Formula: see text]. It is known from Rosenberg’s classification theorem (1965) that the clones [Formula: see text] and [Formula: see text] which consist of all operations on [Formula: see text] that preserve [Formula: see text], respectively, [Formula: see text] are among the maximal clones on [Formula: see text]. In this paper, we find all central relations [Formula: see text] such that the clone [Formula: see text] is a maximal subclone of [Formula: see text], where [Formula: see text] is a fixed central relation.

Unique inclusions of maximal C-clones in maximal clones

C-clones are polymorphism sets of so-called clausal relations, a special type of relations on a finite domain, which first appeared in connection with constraint satisfaction problems in work by Creignou et al. from 2008. We completely describe the relationship regarding set inclusion between maximal C-clones and maximal clones. As a main result we obtain that for every maximal C-clone there exists exactly one maximal clone in which it is contained. A precise description of this unique maximal clone, as well as a corresponding completeness criterion for C-clones is given.

On Some Maximal Clone of Partial Ultrafunctions on a Two-element Set

On Some Maximal Clone of Partial Ultrafunctions on a Two-element Set

English

The structure of the lattice of clones on a finite set has been proven to be very complex. To better understand the top of this lattice, it is important to provide a characterization of submaximal clones in the lattice of clones. It is known that the clones $Pol(\theta)$ and $Pol(\rho)$ (where $\theta$ is a nontrivial equivalence relation on $E_k = \{0,\dots, k-1\}$, and $\rho$ is among the six types of relations which characterize maximal clones) are maximal clones. In this paper, we provide a classification of relations (of Rosenberg's List) on $E_k$ such that the clone $Pol(\theta) \cap Pol(\rho)$ is maximal in $Pol(\theta)$.

Meet-irreducible submaximal clones determined by nontrivial equivalence relations

Let θ and ρ be a nontrivial equivalence relation and a binary relation on a finite set A, respectively. It is known from Rosenberg’s classification theorem (1965) that the clone Pol θ, which consists of all operations on A that preserve θ, is among the maximal clones on A. In the present paper, we find all binary relations ρ such that the clone Pol ρ is a meet-irreducible maximal subclone of Pol θ.

Binary central relations and submaximal clones determined by nontrivial equivalence relations

Let k ≥ 3, θ a nontrivial equivalence relation on E k = {0, . . . ,k – 1}, and ρ a binary central relation on E k (a reflexive graph with a vertex having E k as its neighborhood). It is known that the clones Pol θ and Pol ρ (of operations on E k preserving θ and ρ, respectively) are maximal clones; i.e., covered by the largest clone in the inclusion-ordered lattice of clones on E k . In this paper, we give the classification of all binary central relations ρ on E k such that the clone Pol θ ∩ Pol ρ is maximal in Pol θ.

On sets related to maximal clones

For an arbitrary h-ary relation we are interested to express n-clone Pol n in terms of some subsets of the set of all n-ary operations O n (A) on a nite set A, which are in general not clones but we can obtain Pol n from these sets by using intersection and union. Therefore we specify the concept a function preserves a relation and moreover, we study the properties of this new concept and the connection between these sets and Pol n . Particularly we study R n;k;b for arbitrary partial order relations, equivalence relations and

Конструкция максимального клона точечных функций на полурешётке интервалов

Конструкция максимального клона точечных функций на полурешётке интервалов

TERM EQUATION SATISFIABILITY OVER FINITE ALGEBRAS

We study the computational complexity of the satisfiability problem of an equation between terms over a finite algebra (TERM-SAT). We describe many classes of algebras where the complexity of TERM-SAT is determined by the clone of term operations. We classify the complexity for algebras generating maximal clones. Using this classification we describe many of algebras where TERM-SAT is NP-complete. We classify the situation for clones which are generated by an order or a permutation relation. We introduce the concept of semiaffine algebras and show polynomial-time algorithms which solve the satisfiability problem for them.

On the role of logical connectives for primality and functional completeness of algebras of logics

We consider certain finite universal algebras arising from algebraic semantics in implicational logics. They contain a binary operation -> and two constants 0 and 1 satisfying the axioms 0->x=x->1=x->x=1 and 1->x=x valid in most implicational logics. We characterize the completeness (also called primality) of such algebras, i.e. the property that every finitary operation on their universe is a term operation of the algebra (in other words, it is a composition of the basic operations of the algebra). Using clone theory and the knowledge of maximal clones we describe completeness (functional completeness) in terms of nonpreservation of three types of specific relations. If -> has a simple property and the algebra contains a binary operation @? with a neutral element 1 and a unary operation @? satisfying @?(1)=0 and @?x=1 otherwise, the algebra is functionally complete.

On the (un)decidability of a near-unanimity term

We investigate the near-unanimity problem: given a finite algebra, decide if it has a near-unanimity term of finite arity. We prove that it is undecidable of a finite algebra if it has a partial near-unanimity term on its underlying set excluding two fixed elements. On the other hand, based on Rosenberg’s characterization of maximal clones, we present partial results towards proving the decidability of the general problem.

Maximal clones on uncountable sets that include all permutations

We first determine the maximal clones on a set X of infinite regular cardinality κ which contain all permutations but not all unary functions, extending a result of Heindorf’s for countably infinite X. If κ is countably infinite or weakly compact, this yields a list of all maximal clones containing the permutations, since in that case the maximal clones above the unary functions are known. We then generalize a result of Gavrilov’s to obtain on all infinite X a list of all maximal submonoids of the monoid of unary functions which contain the permutations.

Clones containing all almost unary functions

Let X be an infinite set of regular cardinality. We determine all clones on X which contain all almost unary functions. It turns out that independently of the size of X, these clones form a countably infinite descending chain. Moreover, all such clones are finitely generated over the unary functions. In particular, we obtain an explicit description of the only maximal clone in this part of the clone lattice. This is especially interesting if X is countably infinite, in which case it is known that such a simple description cannot be obtained for the second maximal clone over the unary functions.

Completeness for concrete near-rings

In this paper a completeness criterion for near-rings over a finite group is derived using techniques from clone theory. The relationship between near-rings and clones containing the group operations of the underlying group shows that the unary parts of such clones correspond precisely to near-rings containing the identity function. Rosenberg's characterization of maximal clones is then applied to describe maximal near-rings containing the identity map, while maximal near-rings not containing the identity are described using typical near-ring methods. This finally provides us with a completeness criterion. We apply this criterion to show that if the order of Γ is large then with high probability the set containing a single bijection is complete.

The maximal clones on countable sets that include all permutations

We classify those maximal (also called precomplete) clones on a countably infinite base set that contain all permutations. By a result of Gavrilov's, two of them include all unary functions. The others can be represented in the form $ Pol(mathcal{M}) $ for countably many monoids $ \mathcal{M}$ of unary functions. We give an explicit description of these monoids.

Maximal Partial Clones of 4-valued Logic

Maximal Partial Clones of 4-valued Logic

Classification of the Maximal Clones of Two-Valued Logic

Classification of the Maximal Clones of Two-Valued Logic