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# CATEGORICAL PROPERTIES OF SEQUENTIALLY DENSE MONOMORPHISMS OF SEMIGROUP ACTS

(Impact Factor: 0.62). 05/1388; 15(2):22-23.

ABSTRACT

Let M be a class of (mono)morphisms in a category A. To study mathematical notions, such as injectivity, tensor products, flatness, one needs to have some categorical and algebraic information about the pair (A,M). In this paper we take A to be the category Act-S of S-acts, for a semigroup S, and M d to be the class of sequentially dense monomor-phisms (of interests to computer scientists, too) and study the categori-cal properties, such as limits and colimits, of the pair (A,M). Injectivity with respect to this class of monomorphisms have been studied by Giuli, Ebrahimi, and the authors and got information about injectivity relative to monomorphisms.

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Available from: Leila Shahbaz, Oct 09, 2015
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• "We show that the idempotency and weakly hereditariness of a closure operator C are sufficient, but not necessary, conditions for the well-behavedness of M d -injectivity. Some of these results generalize some of the results in [8], [11], [12], [14], [15], and [16]. "
##### Article: C-ESSENTIALNESS AND WELL-BEHAVEDNESS OF C-INJECTIVITY IN Act-S
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ABSTRACT: An important notion related to injectivity with respect to monomorphisms or any other class M of morphisms in a category A is essentialness. In this paper, taking A to be the category of right acts over a semigroup S, C to be an arbitrary clo-sure operator in the category Act-S, and M d to be the class of C-dense monomorphisms resulting from a closure operator C, we study the properties of M d -essential monomor-phisms and we show the existence of a maximal M d -essential extension for any given act. Finally, the behavior of M d -injectivity in the sense that the three so called Well-behavedness propositions hold is studied. We show that the idempotency and weak hereditariness of a closure operator C are sufficient, but not necessary, conditions for the well-behavedness of M d -injectivity. The class of sequentially dense monomorphisms resulting from a special closure operator (sequential closure operator) and injectivity with respect to this class of monomorphisms have been studied by Giuli, Ebrahimi, Mahmoudi, Moghaddasi, and the author. Some of these results generalize some of the results about the class of sequentially dense monomorphisms.
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• "We show that the idempotency and weakly hereditariness of a closure operator C are sufficient, but not necessary, conditions for the well-behavedness of M d -injectivity. Some of these results generalize some of the results in [8], [11], [12], [14], [15], and [16]. "
##### Article: Well-behaviour of C-dense injectivity of semigroup acts for a closure operator C
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ABSTRACT: Banaschewski defines and gives sufficient conditions on a category A and a subclass M of its monomorphisms under which M-injectivity well-behaves with respect to the notions such as M-absolute retract and M-essentialness. In this paper, taking A to be the category of acts over a semigroup S and Md to be the class of C-dense monomorphisms resulting from a clo- sure operator C, we study the behaviour of Md-injectivity (also called C-dense injectivity or C-injectivity) in the sense that the three so called Well-behaviour propositions hold. We show, among other things, that the idempotency and weak hereditariness of a closure operator C are suf- ficient, but not necessary, conditions for the well-behaviour of C-injecti- vity.
• ##### Article: Injectivity in a category: an overview on well behavior theorems
01/2009; 26(4).