TY - JOUR
ID - 38548
TI - $mathcal{R}L$- valued $f$-ring homomorphisms and lattice-valued maps
JO - Categories and General Algebraic Structures with Applications
JA - CGASA
LA - en
SN - 2345-5853
AU - Karimi Feizabadi, Abolghasem
AU - Estaji, Ali Akbar
AU - Emamverdi, Batool
AD - Department of Mathematics, Gorgan Branch, Islamic Azad University, Gorgan, Iran.
AD - Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.
Y1 - 2017
PY - 2017
VL - 7
IS - Special Issue on the Occasion of Banaschewski's 90th Birthday (II)
SP - 141
EP - 163
KW - frame
KW - cozero lattice-valued map
KW - strong $f$-ring
KW - interval projection
KW - bounded
KW - continuous
KW - $mathbb{Q}$-compatible
KW - coz-compatible
DO -
N2 - In this paper, for each {it lattice-valued map} $Arightarrow L$ with some properties, a ring representation $Arightarrow mathcal{R}L$ is constructed. This representation is denoted by $tau_c$ which is an $f$-ring homomorphism and a $mathbb Q$-linear map, where its index $c$, mentions to a lattice-valued map. We use the notation $delta_{pq}^{a}=(a -p)^{+}wedge (q-a)^{+}$, where $p, qin mathbb Q$ and $ain A$, that is nominated as {it interval projection}. To get a well-defined $f$-ring homomorphism $tau_c$, we need such concepts as {it bounded}, {it continuous}, and $mathbb Q$-{it compatible} for $c$, which are defined and some related results are investigated. On the contrary, we present a cozero lattice-valued map $c_{phi}:Arightarrow L $ for each $f$-ring homomorphism $phi: Arightarrow mathcal{R}L$. It is proved that $c_{tau_c}=c^r$ and $tau_{c_{phi}}=phi$, which they make a kind of correspondence relation between ring representations $Arightarrow mathcal{R}L$ and the lattice-valued maps $Arightarrow L$, Where the mapping $c^r:Arightarrow L$ is called a {it realization} of $c$. It is shown that $tau_{c^r}=tau_c$ and $c^{rr}=c^r$. Finally, we describe how $tau_c$ can be a fundamental tool to extend pointfree version of Gelfand duality constructed by B. Banaschewski.
UR - http://cgasa.sbu.ac.ir/article_38548.html
L1 - http://cgasa.sbu.ac.ir/article_38548_d61135e6f18b53e9ac1eb29192263dbc.pdf
ER -