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Abstract We study quasi-**hereditary** endomorphism **algebras** defined over a new class of finite dimensional monomial **algebras** with a special ideal structure. The main result is a uniform formula describing the Ringel duals of these quasi-**hereditary** **algebras**. As special cases, we obtain a Ringel duality formula for a family of strongly quasi- **hereditary** **algebras** arising from a type A configuration of projective lines in a rational, projective surface as recently introduced by Hille and Ploog, for certain Auslander– Dlab–Ringel **algebras**, and for Eiriksson and Sauter’s nilpotent quiver **algebras** when the quiver has no sinks and no sources. We also recover Tan’s result that the Auslander **algebras** of self-injective Nakayama **algebras** are Ringel self-dual.

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weight λ let ∆pλq be the Weyl module of highest weight λ, ∇pλq its contravariant dual, and Lpλq the simple head of ∆pλq. For any finite saturated set π of dominant weights, the full subcategory of rational G-modules whose composition factors are indexed by weights in π is equivalent to a module category Spπq´mod, where Spπq is a finite-dimensional algebra called a generalized Schur algebra [10]. The algebra Spπq is quasi-**hereditary** (in fact a BGG algebra) with standard and costandard modules ∆pλq and ∇pλq respectively. When necessary we will deal with Spπq-modules instead of rational G-modules for a sufficiently large set π.

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Let A be a finite-dimensional K-algebra. We recall the notion of a quasi-**hereditary** algebra. Suppose the irreducible A-modules Lpλq are indexed by a poset Λ. Let P pλq and Ipλq denote the projective cover and injective hull of Lpλq respectively. Let ∆pλq be the maximal quotient of P pλq whose composition factors are among tLpµq | µ ď λu. These are the Weyl or standard modules. Define ∇ pλq (the good or costandard modules) dually. We say that A is quasi-**hereditary** if for all λ P Λ

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∆ = {∆(1), · · · , ∆(n)}. We denote by F(Θ) the full subcategory of ModΛ consisting of modules which have a filtration with factors in Θ, where ModΛ denotes the category of f.g. left modules over Λ and Θ is a set of modules. These modules are said to be Θ-good. The algebra (Λ, E ) is called standardly stratified with respect to the ordering of simple mod- ules if P (i) ∈ F(∆), for all i = 1, · · · , n [7]; if in addition, EndΛ(∆(i)) is a division ring, for all i = 1, · · · , n, then (Λ, E) is quasi-**hereditary** [11].Standardly stratified **algebras** as a generalization of quasi-**hereditary** **algebras** have been studied recently by some authors in various aspects [1] [7] [9] [15].

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do not yield quasi-**hereditary** **algebras**. Let k be a field, and let M = {1, a, e} be a monoid consisting of three elements, with identity element 1, such that the product of any two non- identity elements is e. Then M is abelian, e is an idempotent in M , and a is the unique nonsplit element in M . In particular, the set of nonsplit elements in M is not an ideal. The algebra kM is not quasi-**hereditary**: an easy calculation shows that kM ∼ = kM e × kM(1 − e), and kM e ∼ = k, while kM(1 − e) is a local 2-dimensional algebra (with basis {1 − e, a − e}). In particular, kM is a symmetric non-semisimple k-algebra, hence any non-projective kM -module has infinite projective dimension, and in particular, kM is not quasi-**hereditary**.

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We use similar methods, starting with Kauman's polynomial in its Dubrovnik form [K] , again keeping a close watch on the ring used, to construct **algebras** whose specialisations can be identied with Brauer's **algebras** [B] in a very natural way. Such **algebras** have been studied in more detail by Birman and Wenzl [BW] , and it was from them that we got to hear of Brauer's **algebras**. While their **algebras** are essentially isomorphic to the ones constructed here, our choice of ring and the use of the Dubrovnik variant of the polynomial allow us to make their conjectured connection with Brauer's **algebras** very directly both algebraically and geometrically. For this reason, and also because of the form of recently discovered connections of both polynomials with invariants derived from Lie **algebras** [T] , we feel that the Dubrovnik version, and the version of P which we use, both have a particularly appropriate choice of variables.

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to groups and Lie **algebras** would result with: if G (or L) is abelian, then G 1 = e (L 1 = 0), and if |G| = p 2 (dimension of L = 2), then G 1 =e (L 1 = 0). So if G (or L) is to have length 2 then |G| ≥ 3 (dimension of L ≥ 3). (Such an L is L = <x, y, z> where [x, y] = z and [x, z] =[y, z] =0. Then L 1 = <z> and L 2 = 0 leaving the length of L to be 2). In [6] it is shown that for G to have length t and |G| = p n , then n ≥ 2 t-1 + t -1 leaving the following table:

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In 1996, Imai and Is´eki 1 introduced two classes of abstract **algebras**: BCK-**algebras** and BCI-**algebras**. These **algebras** have been extensively studied since their introduction. In 1983, Hu and Li 2 introduced the notion of a BCH-algebra which is a generalization of the notion of BCK and BCI-**algebras** and studied a few properties of these **algebras**. In 2001, Neggers et al. 3 introduced a new notion, called a Q-algebra and generalized some theorems discussed in BCI/BCK-**algebras**. In 2002, Neggers and Kim 4 introduced a new notion, called a B-algebra, and obtained several results. In 2007, Walendziak 5 introduced a new notion, called a BF-algebra, which is a generalization of B-algebra. In 6, C. B. Kim and H. S. Kim introduced BG-algebra as a generalization of B-algebra. We introduce a new notion, called a BRK-algebra, which is a generalization of BCK/BCI/BCH/Q/QS/BM-**algebras**. The concept of G-part, p-radical, and medial of a BRK-algebra are introduced and studied their properties.

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In this chapter, we will give the definitions of Lie **algebras** and their representations (over the complex numbers, C ). Then we will define the special class of Lie **algebras** called affine (Kac-Moody) Lie **algebras** and an important class of representations associated with affine Lie **algebras** called highest-weight representations. For more details about Lie **algebras** (especially finite dimensional theory), see [14]. A detailed introduction to Kac-Moody **algebras** can be found in [16].

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Although pre-Lie **algebras** are more familiar to mathematicians than symmet- ric brace **algebras**, applications of these **algebras** seem to be more easily given by symmetric brace formulas [3]. It seems more difficult, in general, to phrase these applications in terms of pre-Lie multiplication. Perhaps, this explicit correspondence between symmetric braces and the pre-Lie product will remedy this situation and cause present and future applications to be solved more clearly.

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It turns out that BCL⁺ **algebras** are a very interesting area of research in the theory of algebraic systems in mathematics. In the present paper, the BCL⁺ **algebras** plays a central role in Liu **algebras**, and the semigroup is actually helping one important aspect of Liu **algebras**, so we put some useful definitions and properties into Section 4, Main Results.

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1. Introduction. Boolean **algebras** that are generated by subchains, that is, subsets that are linearly ordered under the Boolean partial order, were introduced in 1939 by Mostowski and Tarski [7] and have been extensively studied since then. Nowadays they are called interval **algebras**. All basic facts about these **algebras** can be found in [6, Section 15]. We remark, at this stage, that a subalgebra of an interval algebra need not be an interval algebra. For instance, one can consider the algebra of ﬁnite and co-ﬁnite subsets of the ﬁrst uncountable cardinal. This leads us to the study of **hereditary** interval **algebras**, that is, those **algebras** of which any subalgebra is an in- terval algebra. The main concern of this note is to shed more light on these **algebras**. This note is organized as follows. Section 2 deals with deﬁnitions. Section 3 is a pre- sentation of the main theorem. In Section 4, some consequences of this theorem are given.

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obstacles in finding the groups of automorphisms for the polynomial and Weyl **algebras** are the Jacobian Conjecture (for polynomials) and the Dixmier Conjecture (for the Weyl **algebras**) that are still open. The Dixmier Conjecture states that every algebra endomorphism of the Weyl algebra A n is an automorphism, [22]. In [10], it is shown that an analogue of the Dixmier Conjecture

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1. Ulam, SM: Problems in Modern Mathematics. Science ed. Wiley, New York (1940) (Chapter VI) 2. Hyers, DH: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 27, 222-224 (1941) 3. Aoki, T: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Jpn. 2, 64-66 (1950) 4. Rassias, TM: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 72, 297-300 (1978) 5. Cho, Y, Saadati, R, Vahidi, J: Approximation of homomorphisms and derivations on non-Archimedean Lie C * -**algebras**

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There exists a functor from the category of unital crossed modules of bare **algebras** A = (∂ : A → B, ⋗, ⋖), to the category of crossed modules of groups. In the finite dimensional case, this is related to the composition of the functors, sending a crossed module of bare **algebras** to a differential crossed module, and sending a differential crossed module to a crossed module of simply connected Lie groups. In order to work with explicit models, and in the infinite dimensional case, we will do all constructions mostly over C[[h]] (except for the initial construction, only done for motivating purposes), making this relation appear less clear Definition 7 (From A to A • : adjoining an identity). Let A be bare algebra. We denote by A • the algebra

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Soft tissue sarcomas (STS) are also one of the most common subsequent cancers following **hereditary** Rb accounting for 12% up to 32% of all second cancers [6,7,16]. In one large cohort study, an increased risk for STS was first observed within 10 years of Rb diagnosis and continued through adult life up to 50 years after Rb, with specific subtypes occurring at similar ages as in the general population [22,23]. Fifty years after radiation treatment for **hereditary** Rb, the cumulative risk of developing a STS was 13.1%, and the cumulative inci- dence for a STS inside the radiation field was higher than outside the field (8.9% vs. 5.1%) [22]. Table 2 pre- sents the incidence and mortality due to STS after Rb in cohort studies of at least 100 **hereditary** Rb survivors.

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A family with hereditary ataxia Med J Malaysia Vol 35 No 2 December 1980 A FAMILY WITH HEREDITARY ATAXIA C T TAN SUMMARY An Indian family with four members having hereditary ataxia was presented The i[.]

Hereditary ovalocytosis in Malays Med J Malaysia Vol 43 No 4 December 1988 Hereditary ovalocytosis in Malays E George FRCPA*, N Mohandas DSc**, G Duraisamy FRCPA**, N Adeeb MRCOG*, Z A Zainuddin MP*,[.]

ABSTRACT: In this paper, the existence and uniqueness of coupled fixed point for mapping having the mixed monotone property in partially ordered metric spaces which endowed with vec[r]