• Communications of the Korean Mathematical Society
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• v.38 no.3
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• pp.725-731
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• 2023

Following the new description of an oriented full transformation on a finite chain given recently by Higgins and Vernitski in [4], in this short note we present a refinement of this description which is extendable to partial transformations and to injective partial transformations.

• Bulletin of the Korean Mathematical Society
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• v.51 no.6
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• pp.1841-1850
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• 2014

It is known that the ranks of the semigroups $\mathcal{SOP}_n$, $\mathcal{SPOP}_n$ and $\mathcal{SSPOP}_n$ (the semigroups of orientation preserving singular self-maps, partial and strictly partial transformations on $X_n={1,2,{\ldots},n}$, respectively) are n, 2n and n + 1, respectively. The idempotent rank, defined as the smallest number of idempotent generating set, of $\mathcal{SOP}_n$ and $\mathcal{SSPOP}_n$ are the same value as the rank, respectively. Idempotent can be seen as a special case (with m = 1) of m-potent. In this paper, we investigate the m-potent ranks, defined as the smallest number of m-potent generating set, of the semigroups $\mathcal{SOP}_n$, $\mathcal{SPOP}_n$ and $\mathcal{SSPOP}_n$. Firstly, we characterize the structure of the minimal generating sets of $\mathcal{SOP}_n$. As applications, we obtain that the number of distinct minimal generating sets is $(n-1)^nn!$. Secondly, we show that, for $1{\leq}m{\leq}n-1$, the m-potent ranks of the semigroups $\mathcal{SOP}_n$ and $\mathcal{SPOP}_n$ are also n and 2n, respectively. Finally, we find that the 2-potent rank of $\mathcal{SSPOP}_n$ is n + 1.

• Communications of the Korean Mathematical Society
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• v.39 no.3
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• pp.643-645
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• 2024

In this note, we aim to correct some of the results presented in [1]. Namely, the statements of Proposition 2.1, Corollary 2.2, Corollary 2.3, Theorem 2.4 and Theorem 2.6, concerning only the monoids 𝓞𝓟_n and 𝓟𝓞𝓟_n, have to exclude transformations of rank two. All other results of [1], as well as those mentioned above but for the monoids 𝓞𝓡_n and 𝓟𝓞𝓡_n, do not require correction.

• (The) Research of the performance art and culture
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• no.36
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• pp.505-530
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• 2018

The purpose of this study is to examine the meaning of learning methods for education to transmit the items designated as intangible cultural heritages focusing on Seoul-gut. Recently, as the Act on the Preservation and Promotion of Intangible Cultural Heritages was promulgated, 'the archetype' instead of 'the prototype' has become highlighted as a crucial axis for transmission. Although there are some controversies over the definition of the archetype among scholars, it is now possible for transmitters to make use of transformations rather freely being freed from strict orientation to the prototype to follow some fixed frame. Examining learning methods used in education to transmit mudang-gut, one of the items designated as intangible cultural heritages, however, this author has found that the prototype is still emphasized or in the center of learning instead. Presenting learning methods employed for Hwanghaedopyeongsansonoreum-gut of a national intangible cultural heritage, Namijanggunsadang-gut of Seoul Special City's intangible cultural heritage, and Bonghwasandodang-gut as examples, this researcher intends to reveal the reality. In the recent situation that education centering around academies is being widely spread to transmit mudang-gut, setting forth the designation of some of the items as intangible cultural heritages, they are publishing articles on newspaper as an advertisement to encourage learning about mudang-gut. Responding to the advertisement, there are more and more shamans intending to learn mudang-gut coming to the society for preserving items designated as intangible cultural heritages. They can, of course, perform mudang-gut on their own but come to learn it as there is no fixed or definite system for it. Even though the concept of the archetype was introduced, as now it is possible to learn the fixed frame through learning about the prototype regarding the item of mudang-gut as an intangible cultural heritage, those involved in shamanism are coming to it more and more. As transmitting the prototype rather deteriorated the liveliness of gut, those involved in shamanism are coming to it to learn about gut thinking that it is where they can learn the basic frame and also acquire more detailed knowledge about shamanism. Therefore, it is needed for the item of mudang-gut designated as an intangible cultural heritage to accept the aspects of change and develop new methods of education to transmit intangible cultural heritages.