• Journal of applied mathematics & informatics
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• v.25 no.1_2
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• pp.17-33
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• 2007

Suppose F is an arbitrary field. Let ${\mid}F{\mid}$ be the number of the elements of F. Let $T_{n}(F)$ be the space of all $n{\times}n$ upper-triangular matrices over F. A map ${\Psi}\;:\;T_{n}(F)\;{\rightarrow}\;T_{n}(F)$ is said to preserve idempotence if $A-{\lambda}B$ is idempotent if and only if ${\Psi}(A)-{\lambda}{\Psi}(B)$ is idempotent for any $A,\;B\;{\in}\;T_{n}(F)$ and ${\lambda}\;{\in}\;F$. It is shown that: when the characteristic of F is not 2, ${\mid}F{\mid}\;>\;3$ and $n\;{\geq}\;3,\;{\Psi}\;:\;T_{n}(F)\;{\rightarrow}\;T_{n}(F)$ is a map preserving idempotence if and only if there exists an invertible matrix $P\;{\in}\;T_{n}(F)$ such that either ${\Phi}(A)\;=\;PAP^{-1}$ for every $A\;{\in}\;T_{n}(F)\;or\;{\Psi}(A)=PJA^{t}JP^{-1}$ for every $P\;{\in}\;T_{n}(F)$, where $J\;=\;{\sum}^{n}_{i-1}\;E_{i,n+1-i}\;and\;E_{ij}$ is the matrix with 1 in the (i,j)th entry and 0 elsewhere.

• Journal of applied mathematics & informatics
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• v.26 no.3_4
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• pp.541-547
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• 2008

Suppose R is an idempotence-diagonalizable ring. Let n and m be two arbitrary positive integers with $n\;{\geq}\;3$. We denote by $M_n(R)$ the ring of all $n{\times}n$ matrices over R. Let ($J_n(R)$) be the additive subgroup of $M_n(R)$ generated additively by all idempotent matrices. Let ($D=J_n(R)$) or $M_n(R)$. In this paper, by using an additive idem potence-preserving result obtained by Coo (see [4]), I characterize (i) the additive preservers of tripotence from D to $M_m(R)$ when 2 and 3 are units of R; (ii) the additive preservers of inverses (respectively, Drazin inverses, group inverses, {1}-inverses, {2}-inverses, {1, 2}-inverses) from $M_n(R)$ to $M_n(R)$ when 2 and 3 are units of R.

• Communications of the Korean Mathematical Society
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• v.10 no.3
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• pp.561-573
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• 1995

A matrix whose entries consist of the symbols +, - and 0 is called a sign pattern matrix. In [1], a graph theoretic characterization of sign idempotent pattern matrices was given. A question was given for the sign patterns which allow idempotence. We characterized the sign patterns which allow idempotence in the sign idempotent pattern matrices.

• Bulletin of the Korean Mathematical Society
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• v.47 no.4
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• pp.715-726
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• 2010

We continue the study of fully idempotent rings initiated by Courter. It is shown that a (semi)prime ring, but not fully idempotent, can be always constructed from any (semi)prime ring. It is shown that the full idempotence is both Morita invariant and a hereditary radical property, obtaining $hs(Mat_n(R))\;=\;Mat_n(hs(R))$ for any ring R where hs(-) means the sum of all fully idempotent ideals. A non-semiprimitive fully idempotent ring with identity is constructed from the Smoktunowicz's simple nil ring. It is proved that the full idempotence is preserved by the classical quotient rings. More properties of fully idempotent rings are examined and necessary examples are found or constructed in the process.

• Bulletin of the Korean Mathematical Society
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• v.47 no.4
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• pp.787-792
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• 2010

Let A and B be some standard operator algebras on complex Banach spaces X and Y, respectively. We give the concrete forms of linear idempotence preserving maps $\Phi\;:\;A\;{\rightarrow}\;B$ on finite-rank operators.

• Korean Journal of Logic
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• v.11 no.2
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• pp.171-197
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• 2008

In this paper we deal with new standard completeness proofs of some systems introduced by Metcalfe and Montagna in [10]. For this, this paper investigates several fuzzy-relevance logics, which can be regarded as neighbors of the R of Relevance with mingle (RM). First, the monoidal uninorm idempotence logic MUIL, which is intended to cope with the tautologies of left-continuous conjunctive idempotent uninorms and their residua, and some schematic extensions of it are introduced as neighbors of RM. The algebraic structures corresponding to them are defined, and standard completeness, completeness on the real unit interval [0, 1], results for them are provided.