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Register a new userFree objects in triangular matrix varieties and quiver algebras over semirings
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Mark Kambites
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We study the free objects in the variety of semigroups and variety of monoids generated by the monoid of all $n times n$ upper triangular matrices over a commutative semiring. We obtain explicit representations of these, as multiplicative subsemigroups of quiver algebras over polynomial semirings. In the $2 times 2$ case this also yields a representation as a subsemigroup of a semidirect product of commutative monoids. In particular, from the case where $n=2$ and the semiring is the tropical semifield, we obtain a representation of the free objects in the monoid and semigroup varieties generated by the bicyclic monoid (or equivalently, by the free monogenic inverse monoid), inside a semidirect product of a commutative monoid acting on a semilattice. We apply these representations to answer several questions, including that of when the given varieties are locally finite.
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The multiplicative semigroup $M_n(F)$ of $ntimes n$ matrices over a field $F$ is well understood, in particular, it is a regular semigroup. This paper considers semigroups of the form $M_n(S)$, where $S$ is a semiring, and the subsemigroups $UT_n(S)$ and $U_n(S)$ of $M_n(S)$ consisting of upper triangular and unitriangular matrices. Our main interest is in the case where $S$ is an idempotent semifield, where we also consider the subsemigroups $UT_n(S^*)$ and $U_n(S^*)$ consisting of those matrices of $UT_n(S)$ and $U_n(S)$ having all elements on and above the leading diagonal non-zero. Our guiding examples of such $S$ are the 2-element Boolean semiring $mathbb{B}$ and the tropical semiring $mathbb{T}$. In the first case, $M_n(mathbb{B})$ is isomorphic to the semigroup of binary relations on an $n$-element set, and in the second, $M_n(mathbb{T})$ is the semigroup of $ntimes n$ tropical matrices. Ilin has proved that for any semiring $R$ and $n>2$, the semigroup $M_n(R)$ is regular if and only if $R$ is a regular ring. We therefore base our investigations for $M_n(S)$ and its subsemigroups on the analogous but weaker concept of being Fountain (formerly, weakly abundant). These notions are determined by the existence and behaviour of idempotent left and right identities for elements, lying in particular equivalence classes. We show that certain subsemigroups of $M_n(S)$, including several generalisations of well-studied monoids of binary relations (Hall relations, reflexive relations, unitriangular Boolean matrices), are Fountain. We give a detailed study of a family of Fountain semigroups arising in this way that has particularly interesting and unusual properties.
Let $T=left( begin{array}{cc} R & M 0 & S end{array} right) $ be a triangular matrix ring with $R$ and $S$ rings and $_RM_S$ an $R$-$S$-bimodule. We describe Gorenstein projective modules over $T$. In particular, we refine a result of Enochs, Cort{e}s-Izurdiaga and Torrecillas [Gorenstein conditions over triangular matrix rings, J. Pure Appl. Algebra 218 (2014), no. 8, 1544-1554]. Also, we consider when the recollement of $mathbb{D}^b(T{text-} Mod)$ restricts to a recollement of its subcategory $mathbb{D}^b(T{text-} Mod)_{fgp}$ consisting of complexes with finite Gorenstein projective dimension. As applications, we obtain recollements of the stable category $underline{T{text-} GProj}$ and recollements of the Gorenstein defect category $mathbb{D}_{def}(T{text-} Mod)$.
Let $T=biggl(begin{matrix} A&0 U&B end{matrix}biggr)$ be a formal triangular matrix ring, where $A$ and $B$ are rings and $U$ is a $(B, A)$-bimodule. We prove that: (1) If $U_A$ and $_B U$ have finite flat dimensions, then a left $T$-module $biggl(begin{matrix} M_1 M_2end{matrix}biggr)_{varphi^M}$ is Ding projective if and only if $M_1$ and $M_2/{rm im}(varphi^M)$ are Ding projective and the morphism $varphi^M$ is a monomorphism. (2) If $T$ is a right coherent ring, $_{B}U$ has finite flat dimension, $U_{A}$ is finitely presented and has finite projective or $FP$-injective dimension, then a right $T$-module $(W_{1}, W_{2})_{varphi_{W}}$ is Ding injective if and only if $W_{1}$ and $ker(widetilde{{varphi_{W}}})$ are Ding injective and the morphism $widetilde{{varphi_{W}}}$ is an epimorphism. As a consequence, we describe Ding projective and Ding injective dimensions of a $T$-module.
We characterize derivations and 2-local derivations from $M_{n}(mathcal{A})$ into $M_{n}(mathcal{M})$, $n ge 2$, where $mathcal{A}$ is a unital algebra over $mathbb{C}$ and $mathcal{M}$ is a unital $mathcal{A}$-bimodule. We show that every derivation $D: M_{n}(mathcal{A}) to M_{n}(mathcal{M})$, $n ge 2,$ is the sum of an inner derivation and a derivation induced by a derivation from $mathcal{A}$ to $mathcal{M}$. We say that $mathcal{A}$ commutes with $mathcal{M}$ if $am=ma$ for every $ainmathcal{A}$ and $minmathcal{M}$. If $mathcal{A}$ commutes with $mathcal{M}$ we prove that every inner 2-local derivation $D: M_{n}(mathcal{A}) to M_{n}(mathcal{M})$, $n ge 2$, is an inner derivation. In addition, if $mathcal{A}$ is commutative and commutes with $mathcal{M}$, then every 2-local derivation $D: M_{n}(mathcal{A}) to M_{n}(mathcal{M})$, $n ge 2$, is a derivation.
We prove simplicity, and compute $delta$-derivations and symmetric associative forms of algebras in the title.
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