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We study a refinement of the symmetric multiple zeta value, called the$t$-adic symmetric multiple zeta value, by considering its finite truncation.More precisely, two kinds of regularizations (harmonic and shuffle) give twokinds of the $t$-adic symmetric multiple zeta values, thus we introduce twokinds of truncations correspondingly. Then we show that our truncations tend tothe corresponding $t$-adic symmetric multiple zeta values, and satisfy theharmonic and shuffle relations, respectively. This gives a new proof of thedouble shuffle relations for $t$-adic symmetric multiple zeta values, firstproved by Jarossay. In order to prove the shuffle relation, we develop thetheory of truncated $t$-adic symmetric multiple zeta values associated with$2$-colored rooted trees. Finally, we discuss a refinement of Kaneko-Zagier'sconjecture and the $t$-adic symmetric multiple zeta values of Mordell-Tornheimtype.

Truncated $t$-adic symmetric multiple zeta values and double shuffle relations